The first thing I thought when I saw the title of this thread was London Belongs to Me (I've been trying to get a copy of that for a while but every time I see it there is something wrong with it - creased cover, rumpled pages etc - and at £9.99 I want an A1 perfect copy!).
You might like to try Call for the Dead, by John le Carré. Yes, it is a spy thriller, but it's set in and around London in the early fifties and has the kind of feel I think you may be after.
Also, both The Day of the Triffids and The Kraken Wakes by John Wyndham have large segments set in and around London and although Triffids is slightly timeless in it’s setting, they both have a distinct 50s flavour to them.
Lastly, although this doesn't have much set in London, it does do the drab wartime pitch down to a tee, I would recommend Enigma by Robert Harris. Another spy thriller, but brilliantly written and a very good story to boot!