Hi, I'd like to recommend this as a reasonably local and very charming attraction just off the other side of the A205. We went there today for the first time ever as an alternative to the usually busy Greenwich Park ( despite living in south London for years) and were very surprised to see and experience something so gorgeous without hordes of other people being there.
The cafe is on a hillside with great views and good food. The paths are of various difficulties but mostly accessible to buggies and wheelchairs, and just for anyone really. There were people in both manual and electric wheelchairs there today along with others with other kinds of mobility challenges.
It felt miles out of London yet was only a 20 minute drive from Forest Hill.
Highly recommended.