Not in sydenham but the following are all about 10 min drive.
KO kickboxing south London:
http://ko-muaythai.com/news/ko-gym-south-london.html
I currently train here
http://www.scorpionsthaiboxing.co.uk/
Nice club, but a little pricey
http://www.new-wave-academy.com/ ) (in crystal palace, trained by Nick Ocipzak who was in the UFC)
Great club but again a little pricey
http://www.law-wrestling.org/about/london.html also 10 mins away.
only trained here once but seen enough to make me thing its good.
I also do Judo but that's over in Blackheath (where I used to live) its about 20 mins in the car, can give you more details if you are interested.
http://miguelsboxinggym.com/
Its good tried the boxing for a while but ultimately relaised I prefer muay thai, but a nice gym nonetheless.
There is also a boxing gym, in Crystal Palace run by Duke McKenzie, but its very expensive so I've never actually trained there.