The Latchmere is on the 344 bus route. It is crawling with toddlers (apt metaphor!).
I got one of those baby seat things like a rubber ring which the baby sits in, which was great. It meant I could swim and push the baby ahead of me, which the baby thought was funny.
The nicest place in summer is Tooting Lido. A bit of a trek but as well as the Lido there is a big grassy area for picnics, and a childrens paddling pool and a cafe. Arrive early though on sunny weekends.
Brixton Lido was also nice, though from memory it only had a plastic blow up paddling pool and the water never looked too clean.
Charlton Lido is another nice open air pool only open in summer. The baby splash pool sits next to the main pool so its length is equivalent to the width of the big pool. You can sunbathe and there's a park style cafe/snack bar inside. Lovely when it's hot though wasps can be a problem due to sticky ice lolly wrappers in the bins.