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DD goes to the Leisure centre on Latimer Road. She is 4 and has been going for a year. We were on holidays last week and I was pleasantly surprised to see how comfortable she is in the water, swimming under water, jumping in the pool, etc. At her age that's all I expect really . It's very affordable too, so would definitely recommend.
For a two year old you'll need to get in the water too, which is why we waited a bit longer for her as with a younger sibling, the logistics were too complicated.
We go to Dolphis swim school in mitcham, about 5 mins drive from Wimbledon. DD has gone since 6 months and is now just turned 3 and can swim by herself. Really small groups and lovely teachers. Much warmer pool than normal, so great for the little ones.
You have to book for a couple of months, but lots of options on time and days for each age range.