Paul, I live in SE London and there ain't nothink mate. As Gerry says, a few of the parks have ponds that can be fished. Burgess Park off the Old Kent Road has one that has some decent Carp & Tench in it. Also there is huge Crucians in there. Last year I was watching a man pulling Bream after Bream out of the Serpentine in Hyde Park. They were all around the 3lb mark. He was float fishing with bread about 2 rod legths out in 11ft of water. Crystal Palace Park has a lovely little lake in it. (not the boating lake!) Its season ticket only, and theres a waiting list. But if you live nearby it might be worth making some enquiries.
Having said all this, these places are not my cup 'o tea. I prefer to get away from the hustle & bustle of London and fish somewhere more tranquil, which means travelling a fair distance. All the surrounding counties have plenty of waters to try. Day and Season ticket. You should have a look at the RMC Angling website. Most of their waters are within easy reach from London. Also Bury Hill Fisheries near Dorking is definitley worth a visit. Big bags of Tench, Bream and Crucians at this time of year.
Good luck, hope you find somewhere that suits you