Never stayed there but there is a two camp sites on the A21 at Flimwell, Sussex. On one side you have Bewl water which people cycle round and on the otherside there is Bedegbury Pinetum with cycle routes there. Also again never been there, but the Camping and Caravanning Club site at Crowborough is meant to be lovely, and that is a short distance from Ashdown Forest.
I think Senlac Wood might be suitable, though it's a long time since I've been there. Hope this helps.
This is where we had our first camping experience as a family all of 12 years ago around about now. Oh how time flies, a lot of changes since then as well, including an additional child.