I used a hammock a few years ago at (what I think was) Hollows Farm, Grange, Borrowdale, NY 248 169 GB Grid
The site had a little shower/toilet block, was busy, but not like some. It was a lovely location. No Fires. For that, we went down the road (south) to Stonethwaite. I tried to find two trees for a hammock there, but there really wasn't anything suitable, but they did allow fires. Actually, even when we were at the Stonethwaite site we didn't have a fire. Fuel was scarce and after a full day on the hill we didn't have the energy or enthusiasm for hiking back up onto the hill side to try and find some vaguely dry wood.
The way that I look at it, much as I like to sleep in my hammock, in the Lake District, the only way a hammock works is if you are wild/stealth camping on one of the many wooded hillsides. The flat places, where there are usually campsites, are all in fields in the valleys.
We went to five sites, and looked at several more back in 2005. I was able to pitch a hammock at Hollows farm, and there was one suitable spot at a place on the north east side of Buttermere. Otherwise, I was under the tarp on the ground for the entire trip, and on the busier campsites I often wished that I could arrange a little more privacy
Very best of luck with your search!