I meant to attached this to another thread, but… well it's probably better on its own anyway.
The local clubs, for ongoing swimming/training, rather than one-off sessions, are:
Wandsworth Swimming Club
Leander Swimming Club and
Battersea Jaculators (aka Battersea Jacs)
Wandsworth swim at Putney Leisure Centre.
Leander swim at a number of sites including Enrest Bevin College, Tooting Leisure Centre, Balham Leisure Centre, Clapham Leisure Centre… depending on which level/squad you are assigned to.
Battersea Jacs swim at Tooting Leisure Centre.
If you want to go a little further afield:
Chelsea and Westminster (various sites north of the river)
Merton Swordfish (Wimbledon Leisure Centre and Morden swimming pool)
Wimbledon and Merton (?)
Wimbledon Dolphins (Morden pool, Wimbledon Leisure Centre, Wimbledon Leisure Centre, Wimbledon High School)
Merton Pisces (Morden pool, Wimbledon College)
Streatham Swimming Club (Streatham Leisure Centre)
Cheam Maracuda (Morden pool, Cheam Leisure Centre, Westcroft Leisure Centre…)
Kingston Royals (various sites Kingston/New Maldon)
Dulwich Dolphins (various sites around Dulwich)
Other clubs further afield… (Staines, Guildford, Woking, Enfield, Southwark…)
Several of these clubs are very competitive, and run on a regimented basis (as is probably necessary), others are more relaxed and less competitive (several you will not see at open galas, for example).
To get an impression of a club take a look at its structure and what it expects of members once they move on from "lessons" to "training". If you want a more relaxed atmosphere, then the club looking for 3-4 sessions a week for eight to nine year olds probably isn't the one for you.
If you are looking to enter competitions, in most cases you cannot enter unless a Swim England member, and AFAIK you cannot achieve that without being a club member.
If you just want lessons to get confident and competent in the water, I'm sure all of these clubs can help you out – you don't have to move on to the competitive squads, most kids don't.