OP, when you listed five universities, you put QMUL on your list. If QMUL was good enough then, why isn't it good enough now? Please don't allow the lustre of Cambridge, UCL, KCL, and LSE to cast QMUL into the shade.
For my money, I'd rather study at QMUL than LSE, because I think that you might perhaps have a better student experience at QMUL. I base this view partly on some work which I have done with QMUL, partly on the view of a friend who teaches law at LSE, and partly also on the view of another friend who is a distinguished alumnus of LSE. Neither of those two friends would now recommend people to apply to LSE, because they think that it has undergone some negative changes. But all such judgments are subjective.
Please listen to what katana10000 says - he or she knows what he or she is talking about.