Is it true that Godox flashes are fully compatible with another camera system when the latter is used with a matching Godox trigger/transmitter?
Yes, except for, as tugwilson mentioned, the TT350/V350, where it may or may not be. My TT350-O can do remote TTL/HSS with my XPro-F on my Fuji X100; it cannot with my XPro-O on my GX7 and my XPro-C on my Canon 5Dii. I also could
never remotely zoom my TT350-O from any of my XPro transmitters.
The V480 is an update to the V350 that has eliminated most of the drawbacks of the 350 design to keep the units small and light while only increasing about 1cm in each dimension. But nobody has yet reported if Godox also fixed the cross-brand issue with it. My guess is they probably have, since they fixed everything else and updated it to the X3/V100 color touchscreen UI. But it could be an issue.

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In my specific case: I bought a Godox TT685ii for Nikon, as well as an XPro II trigger/transmitter for Nikon. Alongside a Nikon Z6iii, I also use an Olympus/OM camera as a travel camera. Would the TT685ii work fully and natively as an external flash with the Olympus camera, including TTL and HSS, if I buy a Godox trigger/transmitter for Olympus (such as the Olympus version of the XPro II)?
Yes. Olympus has much better compatibility with Godox than Panasonic and isn't seeing some of the weirder shutter timing issues Panasonic users do. But then, Olympus isn't video centric and doesn't have forced EFCS and hasn't removed mechanical shutters.
My old TT685C, once it was firmware upgraded for cross-brand compatibility (the TT685 II came out after all this and was cross-brand compatible out of the box) works fine in TTL/HSS, with remote group, zoom (if i set it to AU on the TT685), and power control as well as TCM from the Xpro-C on my 5Dii/R100, the Xpro-O on my Panasonic GX-7 (with some HSS banding issues), and the XPro-F on my Fuji X100 (though HSS is a bit harder to actually define with a leaf shutter camera, since my sync speed is 1/1000s

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The flash will even display which "flavor" of transmitter it talked to last:
For an Olympus transmitter unit, you can get anything that's designated "O" with an X in its name and it should work (e.g., X3-O, XPro II-O, X2T-O). But if you also wanted an on-camera or accessory flash, you might want to contemplate the V480-O speedlight (so you can do
on-camera bounce flash ), or the iT30Pro-O accessory flash if you just want something like a pop-up but more powerful. All the Godox system 2.4 GHz flashes have built-in transceivers so they can be used either as on-camera transmitter units or off-camera receiver units.
Neither of these, though, will have the full range of function a dedicated transmitter would have (e.g., remote TCM control (TTL convert to Manual) to lock-in/see a TTL-set power level as an M power setting). And the iT30Pro in particular has a more basic feature set as a transmitter. But it's small and can double as an on-
and off-camera flash.