Last Bastion is pretty much as it seems. It is the last stronghold still resisting Yuri's indoctrination. The Sweeper page also notes that it is surrounded by snow and ice, so that suggests an Arctic location, or perhaps hidden somewhere in northern Russia or Canada. It's probably not Antarctica, given that Yuri claims that region and we know that (paraphrased) "no Allied or Soviet forces set foot in Antarctica after the great battle there", so it's unlikely he got shoved out - and I don't think Foehn would be so brazen as to set up their HQ right under his nose.
...I'm going to guess Greenland. It hasn't got a huge population to begin with, and thus Yuri probably wouldn't devote too much of a military presence here.
Wings of Coronia operate from an aerial base of operations, likely called the Coronia. There's nothing to suggest at present where Haihead operates from, but given their focus on assault it's possible they don't have a fixed main base of operations of their own and just operate from the Last Bastion and the Coronia. Remember that they appear to be fully unified, which is a first for the MO subfactions (we all know how disjointed the Allies are, China had that falling out with Russia, and Scorpion Cell is somewhat different to the other two Epsilon subfactions in origin and probably motivation). Given this, Foehn doesn't have to have three separate base of operations, especially given that we know that Last Bastion is the last terrestrial stronghold - it's referred to as a place as well as a sub faction.
The Nanofiber Sync has this little gem: "During the time of the Conscious Slumber, the Foehn Revolt has heavily focused on enhancing and developing new uses for the state-of-the-art nanotechnology, their founder being the co-inventor of it."
There are a couple of things to look at here. First is that Foehn's founder was one of two inventors of Nanotech. Do we know many pairs of scientists in MO lore? I can only think of one; Siegfried and Einstein. And the fact there's only one founder of Foehn ties into Einstein being dead in this timeline (assuming no brain uploading or other trickery), and Seigfried has already been established as using Einstein's legacy/notes to further his work.
Also, "the Conscious Slumber"; I'd say that is a deliberate period of hibernation to escape Yuri's attention. If there's no rebellion for some time, surely he'll let his guard down a bit, right? And if Volknet was still churning away, there's no reason to assume that R&D, and perhaps even production came to a halt for this period.
From Fin and Alize's profile: "Within the ruins of the lost world, an unexpected event has occurred: one of the scouting parties found a pair of survivors, teenage siblings, who have not yet been corrupted by the mind control." Note the term "lost world". It pretty much suggests that Yuri has managed to activate the Mental Omega, and with the exception of Foehn the entire world is under Yuri's control. Fin and Alize are outright stated to be special cases, so it's unlikely that any Allied or Soviet forces remain active at this time. Also note that they are Haihead's hero team, yet "were taken to the Bastion's site" - further suggesting that Haihead share the Last Bastion's home base of operations.
We also know that early Foehn units (specifically Jackals) make an appearance in Act 1 Epsilon mission "Think Different", when China/Yuri raid the Pacific Front and specifically Kanegawa. That doesn't necessarily mean that Foehn was in the making back then, just that Kanegawa was working on units that would eventually become part of Foehn's arsenal.
Foehn's campaign starts later than any other Act 2 campaign, and the only missions currently seen in the list are a trilogy titled "the origin".
We know that Foehn is of limited numbers, but advanced technology.
True to Speeder's words, there's nothing to suggest Foehn has any goal but to defeat Yuri and remove his psychic domination over the world.
A few more bits of general musing:
Foehn's tech is far more "sci-fi" than the other factions. Nano-tech construction? All of their infantry (possibly barring engineers) appear to be cyborgs (which was admittedly already researched by the Russians. Seeing as the Hoplite has vanished with the end of MO3.0, we can't extend that to the Allies). They've got walking tanks and a ton of sonic and railgun weaponry. I don't think that came about from just one faction working on it. We also know (unless this was Speeder trolling us) that the Gharial was originally revealed as an "unidentified Chinese unit", and the unit profile is suspiciously quiet on any sort of origin. The Harbinger and Future Tank are explicitly linked to Foehn and Kanegawa both. And the Future Tank profile indicates that the Pacific Front and Kanegawa Industries are not a true ruler/servant partnership, and that Kanegawa provides aid to the PF voluntarily. This may well have stopped with the Chinese takeover of Pacific Front land as seen in "The Gardener", which also saw Kanegawa scientists rescued by the Euro Alliance, Steinstech and elements of the PF that disapproved of the alliance with China.
