If you are happy to spend more money, then you should head for the more expensive areas famous for fashion, such as Regent Street, Bond Street and Oxford Street. To take Bond Street as one example, here you will find famous British names like Burberry and Alexander McQueen as well as international ones like Chanel and Dior - you can find a full list here (to get an idea of the fashion retailers in any major London street, just type the street name plus 'shops' in a search engine and you should get a list). You should also visit the iconic department stores like Selfridges (Oxford Stree) or of course Harrods (Brompton Road, Knightsbridge).
For less high priced places, you could go further afield. Some of the markets like Camden Town and Portobello Road have great clothing stalls, and it's also worth looking out for smaller boutiques in places like Notting Hill - there is an interesting list here.