I may be wrong, but I would consider the places you named as places to visit, rather than wild camp. Of course at certain times of the year it's a little quieter but places like Grimes Graves can be like Trafalgar Square at times, and I've seen traffic queues in the New forest when the Ponies wander up the road.
A few winters years back I spent some time in the Forest of Dean, but I camped on a farm site as a base for exploring the beautiful Wye Valley.
I am sure there are places to wild camp there, but I needed somewhere safe to leave my tent because I crossed into Monmouthsire a few times and had a totter round the Welsh Mountains, bivvied out and came back to the forest the next day.
It was cold in the forest but obviously more sheltered than up in the mountains.
January, and most days I met people out on mountain bikes or walking the forest rides.
Met a few of the Rangers who patrol the forest too. They have a culling programme of the numerous Wild Boar that live there. I stumbled across these remains one day.
The Wye Valley and the Forest of Dean are well worth a visit. As to wild camping... local knowledge would be useful otherwise you could find yourself challenged by a Ranger or even a school party on their way to visit some of the strange folk who live in the forest...