You could speak to Rangerman, he is a ranger at Beacon country park which I think is quite near you and bushcrafter friendly, he is a member of BCUK. Other than that if you could try any contacts you may have with scout sites, most "normal" camp sites don't really cater for hammocks.
You need permission to camp even in a hammock and just turning up at a wood and setting up without permission would probably be classed as trespass
, and wouldn't be allowed to be discussed on the forum. You may get some replies from people more local to you with better info, but it ain't that easy to find places to go