I was in the same position a couple of years ago and found 3 ppl schools accessable by public transport from London. Will really depend where in London you are as to which suits you best though. I ended up doing at the Pilot Centre in Denham and was very happy there.
Pilot Centre, Denham
Also see
Stapleford - bit tricky to get to.
and
Various schools at Biggin Hill - involves long and infrequent bus journey.
There may be others, i'm not sure. But i was very happy with the instruction and atmosphere at Denham, though the airlfield is a 20 minute walk from the station which i managed once per week. If i was going every day i'd consider getting a car!