Hi, current 2nd year student.
Like most universities, it will be a 'welcoming and enjoyable place' if you put yourself out there by joining a few societies and generally getting involved in uni life. You are not going to be obligated to do any of that, and that is where I feel a few people run into problems during first year. As I said though, I could imagine this would be very much the same at any university.
Politics society is generally quite active, as well as several other politics/IR-related ones such as Debates and Model UN. I don't have a clue how things work in the US but subject societies in the UK tend to be a 50/50 split between social events (which pretty much have nothing to do with the subject, hence why loads of non-econ students are members of our economics society) and academic/careers events. Sports are big at Bath too, so that's another easy way to meet people.
But you are easily able to skip all of that if you are reasonably outgoing and/or are willing to socialise through your flatmates.
Academically I would say Bath is the best out of the list of universities you have given, but something you should probably consider is that Bath is a very science-based university - i.e. it doesn't have any arts/humanities subjects. Whilst that might seem fine when you are applying, I've noticed that it has meant that we have a very apolitical student body relative to other universities (e.g. there are no political party societies or anything remotely partisan, tbh) so I would say that Bath may not be the best place to be politically active but a good place to study the subject academically. In addition to that, I would assume that if you are intending to return to the US after you graduate, non-academic US employers will probably not have heard of Bath. Being a young, small-ish university, it isn't ranked too impressively on international rankings, but is a top 10 university in UK domestic rankings (this is due to the use of different metrics for international and domestic rankings). In the UK I think it's more of a known-brand with employers, but abroad possibly York may be slightly more well-known.