Regarding Bristol, as for diversity, somewhere between 19 and 31% of students there are international anyway. Of the non-international, 23.1% are BAME. For the city of Bristol itself, about 6% are black , about 4-5% mixed race, and 81% white.
Obviously, Warwick is almost stereotypically committed to having a representation of all the world's cultures...in Coventry. Particularly for this subject. 64% of their Law students identify as BAME and 38% are international. (higher than the university across all subjects, for which 29% are international). They seem curiously actively proud that 70% of their Law students are female which, in turn means they're proud that only 30% of their students are male. The average entry grades achieved by Warwick Law students is low compared to any other high ranking universities that have also been teaching Law for over 50 years. For the city of Coventry itself, about 9% are black, about 3-4% mixed race, and about 74% are white.
Nottingham is harder to get in to for Law than Warwick. That should tell you something when people go on so much about Warwick in general. Somewhere between 14 and 27% at Nottingham are international, so likely to be a lower percentage than at Bristol. I wouldn't rule somewhere out because you don't want to bump in to someone you know. Imagine what tricky situations you'll come across in Law itself. For the city of Nottingham itself, 10% are black, 6% mixed race, 66% white.
Regarding private school students, Bristol does have the most of those (about 26%) but I'd be grateful for that because some of the friendliest people I've ever known went to private school. They have less need to be so competitive against other people you see. Nottingham has about 18% and Warwick has about 19%. Nottingham used to have a MUCH higher percentage (about 32%) and Nottingham was solidly a top 10-20 university in general but it has regrettably become a little less fashionable over the years in general with UK students.