Potential Solution: We don't side with either Zeon or the Feds.
This is just a hypothetical, but what if we took a page from Gundam F91's book and center the story around some ragtag third party? In F91 it was essentially space pirates in the form of the Crossbone Vanguard. Suppose we're playing as something similar? People whose homes (be it a colony or a place on Earth) was hit by war between Zeon and the Feds and were forced into doing whatever it takes to survive in a harsh world/space that's being torn apart by a war. The idea is that we're essentially scavengers. We try to salvage shipwrecks and especially mobile suit parts. We do it to both make money and arm ourselves so that we aren't defenseless. Naturally, neither the Federation nor Zeon take too kindly to people like this potentially learning the secrets of some of their machines and so we'd probably be treated like outlaws or pirates.
In my hypothetical scenario, we have a plausible reason for why no one would start out with a Gundam (because their parts are just that hard to come by for us) but also gives us the potential to acquire one or two of them as time goes on, assuming we survive long enough to reach that point.
Edit: I also think at least one person ought to play as a masked antagonist of some kind. It just wouldn't be Gundam without a Char clone. Hell, I'll gladly volunteer for that part myself.
At this point, to be completely honest, I think we'd be cutting out all the reasons to play a UC-based Gundam game. And actually F91 was set around the Federation, just not including Zeon. The Federation was just really weak and borderline unrecognizable by the time of F91, and there wasn't much of a focus on the fact that the soldiers helping Seabook were from the Federation.
But I digress. The point is that cutting out the conflicts between Zeon (and its derivatives) removes pretty much all of the defiining aspects of everything up until the Late UC era. And the alteration of the timeline needed to make it fit really takes away any incentive to use the UC. If you're choosing to make a game that doesn't directly involve the setting's actual conflicts, then there really just isn't a point in using that setting.
It's several unnecessary leaps in logic to explain why there are no Gundams for the PCs yet, too; Gundams were not at all common. They were all very limited production, if not completely unique. There's no explanation needed for why we don't have any, because it's not a given that we would. We're just not in a situation where we would have any.
But given the sort of setting changes you guys would seem to want, it honestly seems like it would make more sense just to craft an AU whole cloth.