A question I can answer!

East London: Brick Lane Books (my old local), Libreria (small, very beautiful, just off Brick Lane). Further east Hackney Books (local favorite), Burley Fisher (also in Hackney, run by an author).
South. Review Bookshop in Peckham, run by the author Evie Wyld. Bookseller Crow is good too.
Central. Hatchards is beautiful. Foyle's is huge but good. Both have great staff. London Review near the British Museum is famous and for a good reason. They also have the best twitter account or did before Muskination.
West. I don't know as well. Bandit country IMO. All the Daunts are good and tend to be in beautiful old shops. Lutyens & Rubinstein has a great selection. Knowledgeable staff.
North. Owl Bookstore in Kentish Town.

There's some comic book/graphic novel bookshops in Soho that are good too. Gosh! Comics is the best.

If more come to mind, I'll add them. London bookstores/shops are great.