it probably depends whereabouts in London you expect to live - getting out for training rides with clubmates will be a key factor and if you live in south London but join a club based in north London you will rarely get together.
If you want to choose your location based on cyling, then I would recommend living near Richmond Park (eg areas such as Wimbledon, Sheen, Barnes, Kingston, Richmond, etc). Richmond Park is a mecca for London cyclists and provides a great training venue for those times when you cannot get out into the Surrey hills. Plus, when you do have a bit more time, living in south west London makes it easy to get into the Surrey hills (a great training and general riding area). The Surrey League is an active racing scene.
The dominant clubs in the area are Kingston Wheelers and London Dynamo. Both are good, but Kingston Wheelers have a reputation for being more friendly. London Dynamo are a bit of a victim of their own success as they are a huge club and some of their members seem to just join up to be able to pose around in the kit on expensive bikes. But the core of the club is probably just as friendly and inclusive as any other club. Both clubs have a mix of social riders and racers of all ages/categories.
I would chek out both websites and maybe hook up with a few training rides and see which one you fancy.
There are plenty of other clubs in south London - Paragon, Addiscombe, etc. but the above two are possibly the most high profile.
North London may as well be the moon to me.