I go to LSE and, honestly, I don't really feel it's that competitive, at least amongst my friend group, but that might be bc relative to the LSE student body, we're pretty much idiots with no career prospects lol. Most people are just pretty career-centric and care a lot about internship ops, which may or may not be a good environment to surround yourself with, depending on the type of person you are.
I would also take all opinions about student life at LSE with a pinch of salt bc ultimately, LSE is what you make of it. With regards to nightlife, London is pretty sick, though, as expected, cost of living is really expensive, so that would be something worth considering.
With regards to teaching quality, LSE is a bit hit or miss, in the sense that some tutors are amazing, whereas others are terrible- my stats teacher openly admitted that he wasn't too familiar with hypothesis testing - which is kinda A-Level stats...
Honestly, I don't know too much about Warwick, but I don't think you could go wrong with either!
Good luck!!