Upgrade 2: Holo Planner. Requires you are in contact with all 3 Resistance Factions. Adds 1 'Wildcard' Resistance Order slot. Costs 125 Supplies and 2 Power, as well as adding a 40 Supply maintenance cost per month. (200 Supplies)
Allows initiating a Covert Op anytime no Covert Op is underway, rather than only once at the beginning of each month.
Can be staffed by 1 Engineer, reducing Covert Op durations by 33%.
Cost: 80 Supplies (100 Supplies)
Maintenance: 10 Supplies.
Build Time: 12 days. (24 days)
Power: +3 Power. +5 more Power if built on an Exposed Power Coil, for a total of +8.
Upgrade 1: Power Conduit. Adds +2 Power, and a second staffing slot. Costs 80 Supplies and adds 10 Supplies of maintenance per month. (80 Supplies)
Upgrade 2: Elerium Conduit. Adds +6 Power. Costs 150 Supplies and 20 Elerium Crystals, and adds 20 Supplies of maintenance per month. (300 Supplies and 35 Elerium Crystals)
Can be staffed by 1 Engineer to add +5 Power.
In the base game, there's a decent argument for building an early Power Relay, especially if your initial room is in a corner and the Exposed Power Coils are as far from your initial room as possible. In such a case, the fastest Excavation route will involve having five rooms open before you have an Exposed Power Coil dug up, of which your initial Power can only use two; building a Power Relay will let you build a Resistance Comms or the Proving Ground or the like, and if you can spare the Engineer then manning the Power Relay will let you use that final room, and getting a Resistance Comms and/or the Proving Ground and/or the Shadow Chamber up early is pretty important to not suffer a game over.
If instead the closest Exposed Power Coil is two rooms below your initial empty room, or Excavating left or right will have an Exposed Power Coil two rooms down from that spot, an early Power Relay is noticeably more dubious. Regardless of how exactly you intend to use Exposed Power Coils, they're a priority to dig open, and with one that close you've got no 'spare' rooms or just the one -and a Power Relay isn't helpful unless you're going to build or upgrade a Facility. Building a Facility of course requires another empty room, and upgrading is dubious given that most of your essential early Facilities can't be upgraded at all. (Or in the case of the Shadow Chamber, can't be upgraded until much later in a run, making it irrelevant to the question of whether to build an early Power Relay or not)
Still, even with an Exposed Power Coil that close, it is faster to dig out the top row, build a Power Relay in the 'spare' room that results while digging up a room on the way to the Exposed Power Coil, and then build a Facility while digging out the Exposed Power Coil. So it's an option to keep in mind.
In War of the Chosen, building an early Power Relay is a lot more dubious of an idea. You have three Facility's worth of Power to start, still only have two Facilities that are immediate priorities (Though the details have changed), while the Avenger's layout hasn't fundamentally changed. The arrangement I laid out earlier where you end up digging out five rooms before starting to Excavate your first Exposed Power Coil is the only arrangement in which you'll end up with enough surplus empty rooms that building a Power Relay results in being able to build an additional Facility with less waiting -and it'll only be the one Facility by default, rather than potentially two of them.
Furthermore, War of the Chosen usually puts less pressure on you to optimize your strategic planning in the first place. Sabotage-the-Covert-Op and Sabotage-the-Resistance-Order both provide ways to knock back the Avatar Project bar without needing to build the Shadow Chamber or increase your Contacts. You can generate Contacts from Resistance Orders and Covert Ops, instead of Contacts being increased only by Resistance Comms and the occasional Rumor. Rapid Excavation, Heavy Machinery, and to a lesser extent Modular Construction can all result in improved Avenger internal timetables. Resistance Network also reduces the pressure to optimize, since you have more wiggle room to jump to where you need to be once you do have Contacts available. And so on.
You can't count on any given such factor to for-sure appear in any given run, but most War of the Chosen runs end up with at least one of these in play, often multiple. As such, it's not so crucial to have a solid build order for the Avenger's internals to avert a game over; what's dangerously slow to get Contacts or the Shadow Chamber online in the base game is quite likely to be ahead of the curve in War of the Chosen. So the majority of the time, you might as well optimize long-term placement, Power usage, etc, instead of making sacrifices in such realms because you're scared of the Avatar Project.
Anyway, it should be pointed out that upgrading an existing Power Relay with the Power Conduit upgrade is actually strictly inferior to building a new Power Relay if you have space and time to spare, costing the same Supplies but providing one less Power. As such, if you can arrange it, it's generally better to build a second Power Relay before upgrading the first one. You shouldn't forget about the possibility of upgrading early, of course; if you need Power urgently, such as because the Avatar Project bar hit max and you need more Power so you can upgrade a Resistance Comms so you can start contacting a region, then no you can't wait six (or more) days for a Power Relay to complete. But it shouldn't be a mindless default to upgrade.
Crucially, many Facilities require precisely 3 Power to build. Building a second Power Relay can let you build a Resistance Comms, or the Laboratory, or the Infirmary, or the Training Center... if your Power consumption is at max, upgrading a Power Relay will only let you build the Defense Matrix or make a few specific upgrades. (Resistance Ring upgrades, Defense Matrix upgrade, Workshop upgrade, and Infirmary upgrade -two of these only exist in War of the Chosen, too) Unless you're intending to immediately slot in an Engineer of course; for a run with an incredible number of early Engineers, it can be easy to justify upgrading and manning your first Power Relay as a way to get a good chunk more Power on demand. But don't just thoughtlessly make the upgrade in the expectation you'll be able to build a new Facility as a result.
As for the Elerium Conduit upgrade... it's frustratingly bad.
Its first issue is that it requires the Elerium Research be finished to unlock it. As such, you simply can't make the upgrade until you're already pretty far in a run, and that late access is really killer to it. If you could have it unlocked early, when you're still in a hurry to Excavate the Avenger and your small team of Engineers has too many things to do, it would be a way to squeeze more Power out without requiring an empty room or pulling an Engineer from another task, and that would be valuable enough to potentially justify sacrificing a good chunk of Supplies and some Elerium Crystals.
By the time you can get Elerium Conduit, though, it's entirely possible you have multiple rooms sitting empty and more than enough Engineers to get everything actually important done, with the surplus finishing Excavation, not because you need it done, but because they might as well do that instead of idling. In such a context, assigning Engineers to Power Relays is simply far superior to purchasing Elerium Conduit; an Engineer manning a slot provides only 1 less Power than an Elerium Conduit, while not costing a big chunk of Supplies or eating precious Elerium Crystals or raising maintenance costs.
This is especially true in War of the Chosen, where the Elerium Research is gated behind the ADVENT Mec Autopsy, you tend to get more Engineers thanks to Covert Ops, and Rapid Excavation and Heavy Machinery can both significantly shorten the time it takes to reach that late-game state of space crunch being gone and your Engineer total being more than you really need. The only good news for Elerium Conduit is that a Breakthrough exists to halve its costs... but while that's something, it's not remotely enough to make up for its fundamental issues or offset how much worse off it is from War of the Chosen's other changes.
It's also worth pointing out that Elerium Conduit requiring the Elerium Research directly contributes to its Elerium Crystal cost being a problem, as most of your Elerium Crystal expenses are from beam weapons and advanced armors, ie the stuff the Elerium Research leads directly into. As a contrast point, the Psi Lab also requires Elerium Crystals, but if you prioritize psionics it will come online in the phase of the game where you have limited ability to spend Elerium Crystals; in the midgame, it's very normal to run out of Alien Alloys and end up sitting on 20 or more Elerium Crystals, unable to use them because most things that use Elerium Crystals also use Alien Alloys, more Alien Alloys than Elerium Crystals at that, and a bunch of purchases require just Alien Alloys. (And Supplies) So the Psi Lab's Elerium Crystal costs tend to only matter in the sense that, much later in the run, you'll be a bit slower to fully make the transition to beam-tier weapons and advanced armors, whereas the Elerium Conduit upgrade's Elerium Crystal costs are a more immediate problem.
One of the most confusing elements of the Elerium Conduit being bad is that its benefits are flat -that is, what I'd intuitively expect would be for Elerium Conduit to improve the Power provided from each sub-component of Power generation. (ie base generation, the Power gained from Power Conduit, the Power generated by assigning Engineers to the staffing slots, and the Power gained by building on an Exposed Power Coil)
To give an example; a Power Relay in a regular room slot that you don't man, don't upgrade with Power Conduit, but do upgrade with Elerium Conduit will be tripling its Power generation, going from +3 to +9. Imagine if Elerium Conduit had relied on a multiplier to arrive at that number, so that it tripled all other Power generation -a Power Relay on an Exposed Power Coil that built an Elerium Conduit would jump from 8 Power to 24. Power Conduit would suddenly be worth +6 Power. Engineers manning a slot would be adding +15 Power. That would be amazing -probably too amazing, honestly, but even just doubling overall Power production would be a lot more worth considering, allowing you to be more space and Engineer efficient at the cost of some Elerium Crystals (And also Supplies, but at that point it would actually be ahead on Supplies) since you'd straight-up drop building the second Power Relay.
As-is, though, Elerium Conduit's benefits are flat, and tuned so it pretty consistently loses in comparisons. Why buy Elerium Conduit on your first Power Relay when building a second Power Relay will cost less and generate more Power if you assign an Engineer to it?
I really wish Elerium Conduit had been possible to upgrade in from the beginning of the game. That would've much better supported a strategy of broad-but-shallow digging, where a player could choose to commit a non-trivial chunk of Supplies and a bunch of Elerium Crystals to getting their Power situation going faster without sacrificing Engineers to the Power altar. As-is, the Elerium Conduit upgrade is largely a trap choice, something for learning players to unknowingly damage their runs by thinking it must surely be worth buying, else why would it exist?
I don't so much mind War of the Chosen not shunting Elerium Conduits down to basic-ness. I'm fairly confident War of the Chosen was consciously embracing 'always beeline for Exposed Power Coils' as the backbone of Facility tuning, and making Elerium Power Conduit a more basic upgrade would almost certainly have ended up at odds with this. War of the Chosen is also pretty light on direct re-tunings of existing content if it isn't tied directly into new content, as apparently something of a philosophical preference of the team, which is not unreasonable.
But still, I don't get why the base game tuned Elerum Power Conduit this way.

WorkshopCost: 250 Supplies. (300 Supplies)
Maintenance: 35 Supplies.
Build Time: 20 days. (40 days)
Power: -3.
Upgrade: Adds a second staffing slot. Costs 150 Supplies and 2 Power, and adds 40 Supplies of maintenance per month. (175 Supplies)
Can be staffed by 1 Engineer, which creates two 'Workshop Gremlins'. These are effectively Engineers, except they can only be assigned to the tiles orthogonally adjacent to the Workshop itself.
I like the idea of the Workshop, but it's suuuuch junk in execution.
On paper, it seems like a sensible enough idea: Engineers are a valuable resource parceled out by the game only fairly slowly, with limited input from the player, and building and then upgrading the Workshop is like buying a couple of Engineers 100%-reliably, instead of hoping extras are offered by Rumors, at the HQ, from Guerilla Ops, or from Covert Ops. The ability to reliably cash in Supplies for pseudo-Engineers sounds pretty good, right?
Unfortunately, a cavalcade of design decisions conspire against it.
First of all is the issue that in the base game you have precisely enough Power to build two Facilities without building Power Relays or digging up Exposed Power Coils, and the GTS and AWC are both extremely important to get online as fast as possible. Slipping the Workshop in early is not a realistic plan unless you luck into Hidden Reserves as your initial Continent Bonus (And aren't in Asia, so you can access it off your initial Contacts) or Power from an early Rumor.
Second is the issue that the Workshop is bizarrely expensive in just raw Supplies. It's literally cheaper to buy an Engineer from HQ, when buying an Engineer won't use up Power, doesn't eat a room slot, doesn't include a maintenance cost, and the resulting Engineer can be freely assigned anywhere, unlike Workshop Gremlins. If the Workshop had been, say, 80 Supplies, it would still be very difficult to justify building it, but it would be a lot easier for stuff to happen like a run gets no early Engineer Rumors but does get an early Power Rumor, and the player goes 'okay, I'll build the Workshop since I have the Power and really need more Engineers'. As-is, only a Legend run might be sitting on enough unused Supplies to be able to justify the cost, and even there it's shaky.
Third is the intersection of the physical layout of the Avenger and the consideration of Engineer-to-Facility assignment rates; only Power Relays, Resistance Comms, and the Workshop itself allow more than 1 Engineer to be assigned, and only two rooms on the Avenger have four rooms orthogonally adjacent to them. The GTS, Shadow Chamber, and Laboratory -as well as the Training Center in War of the Chosen- don't accept Engineers at all. Taken altogether, it's surprisingly difficult to place the Workshop so you can actually use every Workshop Gremlin from an upgraded Workshop -and given how its costs are front-loaded, you really ought to be planning around the idea of maximizing its upgraded form.
Fourth, in the base game you have Facility priorities that provide little room to fit in a Workshop: you'll build the GTS first, then the AWC, and then as soon as you can you'll build the Shadow Chamber or Proving Ground; getting Power from Hidden Reserves or Rumors won't lead to you fitting in the Workshop, it'll just lead to the Proving Grounds or Shadow Chamber getting built sooner. In conjunction with the prior, it's also very likely that by the time your busy schedule can fit in a Workshop, your Facility placement has already sabotaged the ability to squeeze out maximum value from it!
Fifth, actual Engineers are inherently more flexible than Workshop Gremlins. If circumstances change and you realize you need one or more Engineers for tasks you hadn't planned for, you can just shuffle them around to wherever you need them. (With the qualifier of 'you can't remove Engineers if this would take you below your needed Power or Contact count') Workshop Gremlins are more likely to be out of position for such, where you on paper have enough Engineer-slot-fillers to meet all your needs, but in practice you don't now that your needs have changed.
Sixth, the idea of building it early to accelerate Excavation has the key flaw that time spent waiting for it to build isn't really an acceleration. If you assign an Engineer to man it during construction, it will still take 10 days to build; a single Excavation can be finished in 10 days! Indeed, a room that accepts two Engineers will be dug out in 20 days if you just assign the one Engineer to it; thus, building the Workshop and assigning an Engineer to its construction, followed by assigning both Gremlins to the room, will take exactly as long as if you just had the Engineer handle the Excavation directly! If the room instead accepts three Engineers, assigning the lone Engineer directly will take 30 days, while having the Engineer accelerate the Workshop and then the Gremlins handle the Excavation will take... 25 days total. 5 days saved is something, but that's surprisingly underwhelming, and is just the time element, ignoring that the Workshop is expensive and you could by definition have built some other Facility while the Engineer went on to the Excavation. Put another way, building the Workshop to accelerate Excavation is by definition delaying construction of other, more concretely useful Facilities, like Resistance Comms or the Shadow Chamber.
Seventh, Workshop Gremlins don't count as Engineers for meeting Engineer requirements on buildable gear, nor can they be used to fill an Engineer slot in Covert Ops. This isn't huge, but it is yet another reason why the Workshop isn't as good as it looks on paper.
Eighth, Engineers are pretty consistently worth pursuing when offered by Rumors, Covert Ops, or Guerrilla Op rewards; it's difficult for alternate options to compete, particularly when it comes to Rumors. And in the base game, they're one of the better HQ purchases, returning to the issue of a Workshop outright costing more than an Engineer to purchase. As such, it's very rare for a run to have a serious drought on Engineer offers, especially in War of the Chosen, such that a Workshop might be viably worth turning to out of desperation.
Ninth, in the long haul you'll inevitably achieve Engineer saturation. (Unless you just keep getting them killed in VIP missions and not getting Engineers elsewhere, I guess, but if you're struggling that much you're probably worrying about a game over, not whether you should build a Workshop) The Workshop essentially has an expiration date, where the longer you take to build it the less value you'll pull out of it. By the time you have Supplies in abundance, you're running out of reasons to care about the Workshop! You can at least tear it down once it's worthless, where you can't get rid of excess Engineers, but that's a small consolation.
Tenth is somewhat the converse of the above; that your earliest Engineer gains are much more dramatic boosts in utility than your later ones, long before you've hit saturation per se. Your first Engineer is essential for starting Excavation at all. Your second Engineer lets you Excavate multiple rooms at once, or Excavate individual rooms at double speed, or halve a Facility construction time without entirely stopping Excavation, and so on. Your third Engineer speeds up the slowest Excavations by 50% if you're willing to commit everyone, or can be used elsewhere if you'd rather double the Proving Ground's production rate or the like. Your fourth is only speeding up Excavation by letting you dig up an additional room simultaneously, and probably only slowly, and it's probably better to assign them to the Resistance Ring or Proving Ground. Your fifth Engineer is in the same basic boat. And so on.
As the Workshop is horribly expensive, difficult to justify burning Power on early, and hampered by all the layout issues making it hard to optimize it, it's very unlikely you'll build it before you have at least two Engineers, very possibly three or even four, even if you do something unorthodox like build an early Power Relay or decide to put off weapon upgrades in favor of the Workshop being your final initial Facility. ('Final initial' in the sense of it being the Facility that puts you at your initial Power limit) Which is to say you're already hitting notable diminishing returns, where the ability to fill one or two more Engineer slots is nice, but not really dramatic in its benefits.
As such, below Legend it's an extremely questionable investment I personally classify as a trap choice, depressingly likely to lead to learning players getting into trouble because it's not obvious they shouldn't build this hunk of junk.
On Legend, it holds up a bit better. Its price goes up, but only a little, and while it takes twice as long to build Excavation now takes three times as long; building, rushing out, and then manning the Workshop to fully man Excavating a room that accepts two Engineers gives you a final Excavation time of 50 days, vs the 60 days you'd take if you assigned that one Engineer directly; you actually pull ahead by 10 days! With rooms that take three Engineers, you end up with no-Workshop is 90 days, while rush-then-man-Workshop is 65 days; that's almost a month faster!
Furthermore, on Legend Supplies tend to be in surplus from quite early in a run, and the Supply payouts from Excavation are much larger too; you're a lot more likely to have the Supplies to actually afford a Workshop early in the run, and if you endeavor to set up the Workshop to be able to Excavate three rooms it may well recoup its initial Supply cost outright.
I don't usually build it myself on Legend runs, but in War of the Chosen I wouldn't try to argue that's objectively optimal. It subtly helps that the Resistance Ring is a priority to build and to staff; this makes it that bit more likely you'll be able to build the Workshop in a spot where you can maximize the value of its Gremlins immediately and in the long haul.
... it's probably still junk in base game Legend runs, though, unfortunately. Still, it's impressive War of the Chosen made it more viable, given it made no direct change to it and the Workshop Breakthrough only helps with the expense issue, and quite unreliably at that.

Laboratory
Cost: 150 Supplies. (175 Supplies)
Maintenance: 35 Supplies.
Build Time: 20 days. (40 days)
Power: -3 Power.
Accelerates Research speed by 20%. In War of the Chosen, also causes Breakthroughs to be offered more regularly.
Upgrade: Adds a second staffing slot. Costs 125 Supplies and 3 Power, as well as adding 40 Supplies of maintenance per month. (175 Supplies)
Can be staffed by 1 Scientist, adding one more Scientist worth of research speed improvement.
The Laboratory doesn't actually have any requirements, but is rarely worth bothering to build early anyway.
Part of this is that you initially don't have Scientists at all to slap into it, and that its intrinsic Research boost is only really notable once you've got enough Scientists additional Scientists barely affect research speed. Most of it, though, is that XCOM 2 is unusual for a game with a research mechanic in that research boosting isn't actually super-significant. Typically in such games, research bonuses are the 'god stat', such as how often in 4X games factional research boosts overshadow most other possible bonuses; whatever you actually want, research bonuses will probably cover that and a bunch of other stuff too.
In XCOM 2, though, the vast majority of researches have no benefits beyond unlocking purchases and maybe other researches. That is, researching Magnetic Weapons provides zero benefit until you actually buy one of the weapons it unlocked. The exceptions to this rule don't really help; the fact that you need to complete a research to be able to slot in Weapon Attachments is close to a technicality, as Modular Weapons is placed at the very beginning of the game and takes nearly no time at all. You're not going to build a Laboratory to hurry it up; it'll already be done by the time the Laboratory finishes! Similarly, not only are Breakthroughs exclusive to War of the Chosen, but they're also normally unaffected by research boosting, making them irrelevant in that regard. The fact that it makes Breakthroughs appear more regularly is also a bit dubious given how they're erratic about being worth burning lab time on.
Anyway, the big thing here is a matter of tuning: it's not just that research primarily unlocks purchases, it's that XCOM 2 is designed so that you generally don't have the resources to fully benefit from research. That is, if you prioritize unlocking Predator Armor and all your magnetic weaponry, you'll spend a long time sitting on having several purchases sitting in Engineering, waiting for you to accumulate enough Supplies and/or Alien Alloys to actually be able to buy them all.
This is all without building the Laboratory, keep in mind. Building the Laboratory early will increase the period of time in which you're sitting on purchases you can't actually make, which is not exactly helpful.
It also doesn't help that so much research is Instant-able Autopsies, especially in the base game where a lot of them will hit the Instant threshold in no time flat. The Laboratory doesn't help with that at all, after all. This is less of an issue in War of the Chosen, where you don't casually hit Instant on Autopsies so widely and consistently, but it can still crop up readily through stuff like Sitreps that constrict enemy composition showing up very early.
The one bit that actually helps the Laboratory in a meaningful way is that Shadow Projects do, in fact, benefit from the Laboratory, including staffing it. The Laboratory is thus a good idea to get online when you intend to start working on Shadow Projects in earnest, and generally ought to be built before you get started on the last batch of Shadow Projects at minimum so you spend less time on waiting to be able to launch the endgame.
It's worth noting that the Laboratory is one of the Facilities where building it on an Exposed Power Coil is effectively more Power generation than building a Power Relay there... once upgraded. It's not a particularly great option for doing so, but hey, something to keep in mind.

Resistance Comms
Cost: 110 Supplies (160 Supplies)
Maintenance: 25 Supplies.
Build Time: 16 days. (32 days)
Power: -3.
Upgrade: Adds +1 Contact, and a second staffing slot. Costs 125 Supplies and 4 Power, as well as adding 35 Supply of Maintenance per month. (190 Supplies)
Provides +1 Contact base.
Can be staffed by 1 Engineer to add +2 Contacts. The second staffing slot is worth +4 Contacts, instead.
The Resistance Comms Facility is unlocked by the Resistance Communications Research, which is pretty obvious in play. Less obvious is that the upgrade is also locked by research, specifically the Resistance Radio Research. I suspect most players have no idea of this latter point, given you have to unlock Resistance Comms, and then finish a Resistance Comms before you unlock the upgrade or else the game won't actually announce the upgrade, when it's generally best to grab both contact-oriented Researches close together so you can see Continent Bonuses to plan appropriately. I spent more than 1100 hours of play before events happened to line up so I saw the upgrade announcement!
Anyway, the Resistance Comms is one of the Facilities most heavily impacted by the transition from the base game to War of the Chosen, in spite of War of the Chosen not directly changing it.
In the base game, it's an extremely high priority Facility in most runs, up there with the Shadow Chamber for importance due entirely to the Avatar Project bar: killing your first Codex and hitting the Blacksite both knock back the bar and have little or no special requirements, but past that you'll need more Contacts, or to build the Shadow Chamber and perform Shadow Projects, or get lucky and get a Facility Lead. Of these three, getting Contacts up is overall the best thing to pursue: Avatar Project Facilities accelerate the bar's growth just by existing, you'll need additional Contacts to reach the Forge and Psi Gate regardless, and while it's possible to loot Facility Leads or get Contacts from Rumors, these are both sufficiently uncommon you shouldn't plan around them. (And Leads can't unlock the Forge or Psi Gate missions...)
As such, in the base game Resistance Comms should be one of the first Facilities you build once your Power is expanded -in fact, there's a decent argument for putting off the Advanced Warfare Center in favor of an early Resistance Comms, if your run hasn't lucked into early Power or Contacts from Rumors or the Hidden Reserves Continent Bonus. You need to keep the pressure on the Avatar Project bar to not game over, after all.
Furthermore, a second Resistance Comms can end up necessary, depending on where the Forge and Psi Gate generate, and your general luck with Avatar Project Facility placement. You usually don't need to upgrade both -one fully upgraded and manned Resistance Comms already puts you at 11 Contacts in conjunction with your base 3 Contacts. 12 Contacts may be necessary, but unless you really want all the Continent Bonuses, much more than that is probably a waste.
In War of the Chosen, building and especially upgrading a Resistance Comms is a lot less of a priority. Most runs should build one at some point, but in War of the Chosen you've got a much wider variety of tools relevant to this point, and more reliably get a hold of at least one of these tools: a run that gets Resistance Rising I and/or II doesn't need to rush out a Resistance Comms to get to where it needs to go. A run that gets an early Sabotage Covert Op and uses it has more time to work with. A run that picks up Contacts from Covert Ops can, again, put off building a Resistance Comms. A run with Resistance Network can put off upgrading the Resistance Comms until literally right before the point an additional Contact is actually needed. (And possibly end up never making said upgrade due to acquiring Contacts some other way, or knocking back the Avatar Project some other way)
And a run that gets a hold of none of the new tools can still pick up Contacts from Rumors or luck into an early Facility Lead and decide it's worth cracking open, just like in the base game.
Indeed, as I noted back in the Breakthroughs post, I've had runs that never built a Resistance Comms at all! More typical is only building and upgrading the one Resistance Comms... but I've also had runs that built the Resistance Comms and never got around to upgrading it.
As such, while you shouldn't assume your War of the Chosen run will get to build just the one, you conversely shouldn't particularly assume you'll need a second one, where in the base game you usually will need a second one, to the point you should plan from the beginning around the idea you'll need a second one.
All that said, the Resistance Comms is a Facility you should be getting up sooner rather than later, particularly in the base game. You need Contacts to be able to knock back the Avatar Project more than a handful of times, and it not only takes time to build a Resistance Comms but also time to make contact, as well as to build Radio Relays if you're traveling longer distances -which you will be. Starting Resistance Comms when you've got no spare Contacts, the Avatar Project bar is 2 away from full, and the nearest region providing an option for knocking back the bar is two or more Contacts away is a recipe for a game over. Less so in War of the Chosen, where you might have Resistance Network and/or a Sabotage Covert Op able to swoop to the rescue, but this comes with the qualifier that the Fatigue system means it's still better to have contact with places sooner so you can fit missions into your schedule comfortably, rather than having the game force you to send Tired and/or under-leveled soldiers because all your best soldiers are Tired or Shaken or recovering from injuries and you have to do the mission right now.
As an aside, the Resistance Comms is another Facility that, once upgraded, is more Power gain from putting on an Exposed Power Coil than a Power Relay. So one idea is to simply build your first Resistance Comms on the first Exposed Power Coil you dig up. Again, this isn't a great idea, particularly in War of the Chosen, but it's an option to keep in mind.
Advanced Warfare Center
Cost: 115 Supplies (175 Supplies)
Maintenance: 35 Supplies.
Build Time: 21 days.
Power: -3.
No upgrades.
Accelerates soldier healing by 50%, and allows soldiers to gain bonus skills.
Can be staffed by 1 Engineer, doubling the rate at which non-SPARK soldiers heal. (Relative to having the unmanned AWC up)
Can also be staffed by 1 soldier to retrain them, resetting their skill selections at the end of the training.
This, of course, only exists in the base game, as War of the Chosen splits it off into the Infirmary and Training Center. I've also covered the main of its utility previously, and so have little to say here.
Note that the Advanced Warfare Center first requires you complete the Alien Biotech Research (ie studying the chip in the Commander's head) to unlock it. I'm... not entirely sure why, honestly, but it does mean you can't build it as literally your first Facility? But it's not like you'd want to when the GTS is so much more urgently important...
As the AWC's bonus skills aren't awarded retroactively without using retraining, you'll ideally get this up early. Base XCOM 2 also really expects you to use basically one squad's worth of soldiers continuously, so injuries are very disruptive, and minimizing time spent injured is very helpful unless you're improbably reliable about preventing soldiers from being injured. (Improbable in part due to the base game having several dubious and/or glitchy mechanics that can and will get people shot when you made no apparent mistake) This is especially true if you have Alien Hunters, since Ruler Reactions make it extremely difficult to completely avoid damage when fighting them.
As such, while you can put off the AWC until later and just use Retraining, it's generally better to get it up early. If you want to play the strategic game safer, though, there's an argument for putting it off in favor of the Proving Ground, or the Shadow Chamber, or your first Resistance Comms, so you can get started on knocking back the Avatar Project sooner. Commander difficulty in particular has a sufficiently hostile and RNG-swingy Avatar Project situation that not rushing for tools to knock it back legitimately has potential for you to game over without an opportunity to do anything about it. It's unlikely -among other points, there's ways the RNG can offset such bad luck, like providing an early Facility Lead or Contacts or Power from Rumors- but not so unlikely I'm willing to declare it an ignorable outside chance.
I've never bothered and never gotten a game over as a result, myself, but still. If you're concerned, the AWC being up early isn't so essential you're severely handicapping yourself. (Unlike the Guerilla Tactics School, which is possible to do without -there's an Achievement for beating the game with no squad size expansions purchased, in fact- but the game very much expects you to pick up its benefits ASAP, and you will suffer for the lack)
Training Center
Cost: 125 Supplies. (175 Supplies)
Maintenance: 30 Supplies.
Build Time: 12 days. (24 days)
Power: -3.
No upgrades.
Allows spending Ability Points on core class soldiers, and by extension unlocks access to bonus skills on them.
Can be staffed by 2 Bondmates (Eating 1 slot, with only 1 slot available) to level up their Bond.
Can also be staffed by 1 soldier to retrain them, resetting their skill selections at the end of the training, using a different slot.
I have, of course, covered the main of the Training Center's utility before, in the Training Center and Bonds posts, and so have relatively little to say here.
Unlike the AWC, the Training Center has no requirements: if you feel like it, it can be literally your first Facility in a run. You shouldn't do that, as both things it provides take a bit to really become meaningfully relevant, but it is possible.
Bonds-wise, you don't actually need the Training Center for the first Bond tier, and it tends to take a bit to get anyone to the second tier such that you need the Training Center for further Bond improvements, especially if you're even slightly picky about Bondmates, or are quick to get a SPARK online and aggressive about deploying it so that there's slightly fewer opportunities for Bonds to get going. So no rush here, even with the second Bondmate tier being the biggest payoff.
Skill purchases are more likely to actually be possible to perform with an early Training Center construction, but early in a run you'll have barely any X-COM Ability Points, and your soldiers won't have enough levels under their belt to readily purchase skills entirely internally. The 25-point skills won't be possible to purchase at all so early -even a Savant Sergeant you're willing to dump all your X-COM AP into usually won't be able to- which immediately means some of the highest-value skills like Rapid Fire are not, in fact, going to be accessed amazingly early through the Training Center. There's non-bonus skill purchasing too, of course, but XCOM 2 is pretty decent about the higher ranks tending to be where you find the better skills to buy; it's not remotely perfect at this (Why are Steady Hands and Aim a level above Faceoff, game?), but it's still the case that high-value purchases are largely not found at the lower levels in the first place. It's not until the Lieutenant or Captain ranks that you're really hitting the 'I want both skills at this rank' moments -and on Specialists, it's only really Major and Colonel that provide pairs of skills you'd prefer to not have to pick between.
As such, it actually takes a fair chunk of time to reach the point the Training Center is particularly likely to provide access to skills you'll both want to purchase and be able to purchase.
I started out building the Training Center as my third Facility, but nowadays I usually don't get to it until sometime after my first Exposed Power Coil is dug up; that's roughly around the point it starts really paying off anyway! This frees up the third Facility to be the Proving Ground, which is a lot more consistent about paying off if built early.
InfirmaryCost: 115 Supplies. (175 Supplies)
Maintenance: 35 Supplies.
Build Time: 21 days. (42 days)
Power: -3.
Upgrade: Hyper-Vital Module. Allows you to send an injured or Tired soldier out in mint condition, once per game for each given soldier. This state of good health is only for the duration of the mission. Costs 80 Supplies and 2 Power, and adds 25 Supplies of maintenance per month. (150 Supplies)
Accelerates non-SPARK healing by 50%. Additionally, soldiers will automatically recover from Phobias upon participating in 3-5 missions without suffering an injury.
Can be staffed by 1 Engineer, doubling the rate at which non-SPARKs heal. (Relative to having the unmanned Infirmary up)
Can also be staffed by 1 soldier with a Phobia to remove all their Phobias. This takes 5 days to complete. (10 days on Legend)
The Infirmary inherits the AWC's unlock requirement: completing the Alien Biotech Research. This is a bit unsurprising given it's literally the AWC, visually; that's not me accidentally copying graphics without noticing, that's how it is in-game. Which is efficient, but still a little surprising.
Unlike the AWC, the Infirmary isn't hugely important to get up quickly unless you pick up a lot of Phobias distressingly early. War of the Chosen starts you with more soldiers and forces you to rotate them via the Fatigue system, making injury times much less punishing; improving turnaround on injuries is nice, but not particularly important until you're later in a run.
The Phobia end of things is a bit more complicated, depending in part on skill level, difficulty, and raw luck. As your skill rises, Phobias tend to occur less often, as you go up in difficulty they tend to occur more often, and how important it is to clear Phobias is down heavily to what the RNG elects to even throw at you; if you're just getting the fairly harmless Phobias, the Infirmary isn't important. If some of your best soldiers all pick up seriously problematic Phobias, you're gonna want the Infirmary up ASAP!
Typically, though, the Infirmary is best to put off until later in your run. Its benefits are almost never important early on, and there's too many other Facilities you'll want up early for it to be terribly likely to get fit in by just getting some bonus Power.
A pleasant surprise is that the Infirmary does, in fact, cause Engineers and Scientists to recover from injuries faster, exactly as per soldiers. This isn't a huge thing -Scientists and Engineers can only be injured by Chosen Sabotage- but I'm honestly surprised they weren't overlooked.

Proving Ground
Cost: 100 Supplies. (125 Supplies)
Maintenance: 25 Supplies.
Build Time: 14 days. (28 days)
Power: -3.
No upgrades.
Performs assorted Projects. Only one advances at a time, though you can queue more than one.
Can be staffed by 1 Engineer to halve production time on Projects.
To unlock the Proving Ground, you have to perform the ADVENT Officer Autopsy, as Shen insists the Skulljack can't be made with just your existing tools.