In reply to FLclimber:
Basically there is no climbing close to London.
You could try southern sandstone but it's unusual, top rope only because it's soft, full up (with Londoners) and some places you have to pay to climb.
There's lots of sport climbing in Portland but it is a 3hr drive.
Basically you should go somewhere else, Peak, Wales, Lakes or some of the amazing sea cliff venues, north Devon and Cornwall coast, Pembroke etc etc (or Scotland but that is quite a trek from London).
That's pretty much what Londoners do (or fly to somewhere in Europe for sport climbing).
As for sport or trad - mostly it's trad, mostly it's single pitch, mostly you don't need a double rack of cams. Since you'd have to bring a lot of gear you should probably decide!
This will help:
https://www.ukclimbing.com/logbook/map/
Post edited at 21:19