Sorry, cross posted with Highbinder.
They do shed branches, (that's why the thread really I suppose) but they are all different. Beech is a No-No! because it's a known hazard, but others, like Ash and Sycamore are generally fine (that's the UK Sycamore, not the American one, I mean)….it really depends on the area and what's growing.
Open land here is generally either farmland or moor. If you're camping on a moor expect midges, sodden wetness underfoot, and no shelter if the weather turns (pretty common really on these islands) bad.
We have a lot of tree cover still though, and certainly in Scotland forests are freely accessible. The average is 12% tree cover in the UK as a whole, and mind that covers bare uplands as well as towns, farmlands, etc., Most of us can find woodland.
Personally I don't do hammocks. I do have one and all the accoutrements but I'm become a creature of habit and admit that these days I like my tent
I also admit that I don't read hammock forums
M