The branch that the leaf is connected to is more likly the cause of the problem. But that is nitty picking.
Perhaps the main question that should be asked is what rights of protection do BT, or any other utility for that matter have? Yes they should no doubt be able to claim if you caused wanten damage to their property but do they not have a duty to take reasonable steps to protect it themselves?
I have an overhead elecrticity supply to my property and the roadside lines run at around fence position (allbeit a lot higher!) and I had a fairly large tree in my garden. Every so often the elecrticity utility would call and as if they could trim my tree. By a quqlified person and at no cost to me. So really BT should be doing the same. Or did they and you refused permission?
Regarding the unwanted overhead cable. There are ways of getting it moved, for free, if you go about it the right way!
As an aside do you have a BT pole on your property or just the cable? Could make a BIG difference!