Southwest London has some great parks with forest-like areas. If you want the real thing you'll have to go further afield.

Round my ends there's remnants of the ancient forest which once covered most of what's now London.

Park Wood, Mad Bess Wood and Copse Wood in Ruislip and Bayhurst Wood and Old Park Wood in Harefield. If the weather's good take your bike on the tube, get off at Ruislip Manor and head up the road past the Weatherspoons. You'll see the woods from the top of the hill.

https://en.wikipedia.org/wiki/Ruislip_Woods