Kingston Ridge in Langdon Hills, Basildon, Essex (has running water but showers need a generator to run). It's a general campsite but has a bushcraft area. Lots of hanging locations for hammocks, but not necessarily in the bushcrafting area (I've not been in that area yet!)
King Edward VIII (Riffhams), Danbury, Essex (Running water and one electric shower). There is a tree preservation order on the site (and is also an SSSI) so you're limited to what you can pick off the ground! I've done shelter building there - very good for it. Majority of trees are oak or sweet chesnut, especially good during the autumn. Fires only at designated spots due to possibilities of root fires. Abundance of good trees for hanging, downside is there is a lot of dead wood in the trees as you're not allowed to pull it down.
Belchamps, Hockley, Essex - has a bushcraft area where you can pretty much do what you want as long as you leave no trace. The site as a whole can be crowded at times with kids running through your site. Most expensive option but it has a lot more facilities (lots of activities as well) Only a couple of good hanging locations.