London is a pretty big place and there are plenty of places to train at, so it may be worth asking about places that are close to where you live or if anyone knows of any places near where you live, rather than if anyone can recommend somewhere "near London".
Airfields tend not to be convenient for public transport. Best I can think of is Biggin Hill. Bromley south is the nearest station (15 mins from Victoria) and there is a bus from right outside straight to the airport (320).
Biggin Hill also has 3 or 4 flying schools offering reasonable packages. It's not a cheap place to train though as landing fees can be a bit pricey, but one of the benefits to the place is full ATC and of course there is a very long hard runway.
Best thing to do is look at what's available nearest to you though since the weather can change even while your on the train!