Bunker 51 play every other Saturday night. Expensive and not the best though.
Red 1 Chislehurst will very rarely run a Saturday game.
Very few sites run Saturday games in the whole of the South East, let alone London.
Hockley in Essex can be accessed by public transport and run games Thursday, Friday, Saturdays and Sundays. Please be aware that the FPS limit is only 290 and you need full face protection and full seal goggles (Not mesh) to play.
West of London, Skirmish Wycombe do a game at least one Saturday per month. You didn't say which part of London you're from, so it may be do-able for you.