London is a great place to study philosophy. As I mentioned, I was enrolled in the Philosophy Department of King's College London, where one can study nearly any area of philosophy. For a more comprehensive categorization of specialties one should consult Brian Leiter's Philosophical Gourmet Report, though it is also recommended that one do one's own research.
Apart from King's there are several other philosophy departments of note in London. They include UCL and Birkbeck, as well as the LSE and Heythrop. There is also some interesting cross-disciplinary work going on. For example, check out the Centre of Medical Law and Ethics, headed by Professor Jonathan Glover, which is a program that encorporates Law, Ethics, and other fields of study.
There is also the University of London SAS Institute of Philosophy which brings a series of speakers to London each semester as well as a couple of Visiting Fellows to do research.
And a short train ride away are the departments of philosphy at Oxford and Cambridge.
A bit further afield you can visit Sheffield, Bristol, or Warwick.
This by no means exhausts what is philosophically happening in or near London, but it is a start.