Downham leisure centre does mum and baby classes. There are also private baby lessons held in Charlton (think the company might be called swimkidz)
We do Aquatots in Catford, Bromley Road. It’s pretty good. Warm private pool, not great changing facilities, but friendly and small.
We do Aquatots in Peckham and find them great
Eltham Centre lessons are reasonable, but more about playtime than anything progressive at the age I went (4-9months)
Aquatots are fantastic! More expensive than others but they have a great, progressive curriculum and their pools are much warmer than at leisure centres.