Hi, we used to take our now-12 year old to Clapham Manor about five years ago on a Saturday morning. I don't know if it still going, but the older gymnasts used to be amazing. The same people also ran classes in Brixton. We also know of a couple of children who took gymnastics seriously and used to trek down to somewhere like Camberwell several times a week.
An alternative is Circus Space in Hoxton, a bus ride away from many parts of SE1. They have a full youth programme on Sundays starting at about 5 years old. Tumbling, tightwire, trapeze, juggling etc. Really recommended.
I think you may be a bit last for this term at Circus Space though it does not start till 19 Sept, but ask also about holiday events and taster days. If you do find there on a Sunday do let me know as it would be nice to meet you.