Possibly the lower braches are growing horizontally into your neighbours' garden and encroaching on flowerbeds or lawn, making the area shady and damp. That may be why they wanted the branches trimmed on their side.
I believe there is no legal obligation for the owner of a tree to trim the neighbour's side. However, I do think it would be more neighbourly to get it done, particularly as your neighhbours are in their 80's. Some people, as they age, become concerned about issues that would not have concerned them at a younger age and they become anxious.
The fence panels don't sound very safely placed and possibly look untidy.
As others have said, perhaps it would be a good idea to speak to someone at the Housing Association to see what their views are.
Is it possible to remove a topic from "I'm on"
By special request, let’s discuss our favourite Classic Music and why?
