Scottish Power cut ours back, but never again. They were very helpful, and left us with the wood for our wood burner, but their idea of a pollard was horrible. We coppice our own now. It's only when they get to within a couple of feet of the wires that they get insistent.
We had a giant Christmas tree getting ever closer to some wires - we cut it back once but it never looked right, so we removed that one completely ourselves.
As for who 'should' do it, well, if your trees damage your power line then you pay for the repairs. If it's a larger wire supplying others then I don't know.
As for the lane, usually the branches are kept back by lorries driving through, making a tunnel effect. We have a tree which has branches projecting horizontally towards the road low down, so we will be dealing with them this winter, for safety. If the Council do it, they take the wood.