Hi, current UWE Law LLB student here. I 100% second Bristol as where to go for student & night-life. As well as the train, there's also a National Express coach service between Bristol & Gatwick Airport via Heathrow Airport.
You've got UoB on your list, but I'd also recommend taking a look at UWE as well. There is a large international student community here, particularly in the Law School, and last year I actually lived with a guy from Canada who was on a year abroad. I study at Frenchay campus, which though it is less central than where UoB is based (mainly Clifton), it takes 15-20 minutes to get to Bristol city centre by Metrobus, and the M1 is 24/7 now. The Law school also offer a number of extra-curriculars to boost skills & employability, such as placement years, Pro Bono schemes & skills workshops. If you have any questions about studying Law at UWE, I'd be happy to answer.
In terms of your other choices, I've heard mixed reviews about the University of Leeds, it seems like there's quite a gap between the best and the worst departments, and I don't know where the Law department falls into that. Manchester & Nottingham are great student cities, with the former having a particularly great reputation in the UK for its nightlife. Though I did look at & apply for university in Brighton, where the University of Sussex is based, I didn't get to visit, and it looks to be a rather expensive place to study. Liverpool also seems to be a good student city, but as it's near Manchester I'd probably choose to go there instead lol