Racking my brains, as we have got beyond this stage.
If you live towards the west of the area, I would recommend the gym classes at Clapham Manor Leisure Centre, though they start at 8.30 am on Saturdays. I think there is also gym in Dulwich.
Swimming is something that kids this age do. Mine went first to the Latchmere in Battersea, then to Clapham Manor, though there are also classes at the Queen Mother near Victoria and potentially at the Oasis, or that one at Surrey Quays, or Peckham or Dulwich.
You dont say what sex your kids are but lots of small girls do ballet. Again friends seemed to take their kids to somewhere in Westminster on a Saturday morning.
An amazing local resident has set up a variety of sports including tennis at the old Lilian Baylis School just inside SE11. Sports Action Zone, administed by the South Bank Employers Group should be able to give details.
There is masses of footie, but also tag rugby, though dont know what age that starts. This I think takes place in Dulwich (Old Alleyns) and Tooting (Battersea Ironsides).
If any specific sport takes your fancy why not look it up on the Council website or contact the Council's sports department.
Or PM me with your phone number saying what you are interested in. My kids are now 8 and 10 and so I may know someone who signed their kids up.
Good luck. It does get easier when they get older, though I now feel I am spending too big a portion of my life commuting to South London Swimming Pools.