The UK was probably a part of the European Commonwealth, which dissolved into bickering nation states after a war with the middle east.
Most likely, but that sort of thing probably doesn't make that much difference once a nuclear war happens, no matter how well your country is doing at the time, things are going to be pretty bad afterwards. How bad? We don't know, we really don't have much clear information about the state of the world outside America, so I'd say this is a subject that would give Bethesda lots of room to get creative, if Bethesda choosed to explore it.
Now, it would be nice to see a Fallout game outside of America. I mean, I'm fine with Fallout games set on former American soil, and the series could probably continue to go on for quite some time without running out of new things to show without leaving what was once the U.S. After all, the United States is not a small country, and there's a lot of places that we've yet to see in the Fallout world, so we really don't know what most of the country looks like after the war. But in the end, America is not the only place in the world, and I'd like to see what things are like in other countries, this could allow us to see more about how the nuclear war effected not just America, but the rest of the world as well. It would also give future a chance to try a lot of new ideas without fear of introducing continuity errors, it would be an oportunity both to expand the Fallout universe and to test out new ideas.
Still, I suspect that in the immediate future, Fallout games will continue to be set in America.