ooh i second all of the above ideas:
museum of childhood (where mine crawled for the first time! ),
princess diana playground,
the littlies basement in the science museum,
battersea park (inc zoo - which has ace playground - and you can hire bikes with a bucket seat at the front for children to sit in).
And with summer on the horizon... (ha ha)... how about:
The lido at hampstead heath. It has a baby splash pool - and the playground at the south end of the park, near the lido, has a paddling pool in summer. If you're in east london, the overground may get you there quicker than driving.
The lido at hyde park is great too, that also has a paddling pool and playground and lots of grass for running around. Both the lidos cost, but not a huge amount.
Museum of london not that useful for a toddler but my 5 year old loves it
And we used to go to Coram fields in bloomsbury a lot (we no longer live in london) such a good place in central london to let them burn off steam. It's secure like the princess diana playground - no adults without kids. There's a paddling pool in summer. Two playgrounds for kids of diff ages, mini farm, cafe. It's just generally awesome. And freeeeeee :)
And if you fancy pubgrub before/after, the marquis cornwallis nearby on Marchmont street is fantastic.
Hope this helps!