It was a messy situation on multiple counts. Remember, the Soviet Black Sea Fleet refused to take orders from Ukraine, and iirc it had it's own nuclear weapons. There was also a potential secessionist crisis with Crimea back then, and an ongoing war in Moldova. There was no real way that a Ukraine with a collapsing state socialist economy suddenly turned capitalist was going to be in a position to tell both the west and Russia to kick rocks at the same time.
Personally I don't think the situation would be very different. I think there's no way Ukraine could keep the bombers flying or the ICBMs functioning. They're too complex and expensive. They could probably keep a small inventory of tactical nukes, but even there they'd be under continuous pressure from all parties to shrink or get rid of that arsenal. And they didn't really have the means to make more. It's very likely that even if Ukraine initially didn't give them up, they would have disposed of most of them over time, possibly all of them.