Crikey - this is clearly a subject, like parking, which arouses strong emotions! The best way of dealing with this, if you can is to have a chat with your neighbour, point out the issues and ask how you can resolve it between you. Be sensitive OP because as you can tell, as soon as you start talking to a neighbour about something which is theirs, on their land, all some people will hear is 'bloody nerve, wanting to chop down my tree'. Wowwow's first comment was right - in the UK, providing there isn't a TPO, you can cut off the overhanging branches from your garden.
I completely digress, but we once had a neighbour who wanted to replace the 6 foot fence in their garden, their side of the boundary wall in order to 'straighten it up and make their garden bigger'. How did he think he was going to achieve this? By making my garden smaller! He couldn't see the original flag wall behind the fence which actually marked the boundary (they thought their fence was the boundary) and he actually stood in our garden one evening, pulled at the flag wall (completely immoveable btw) and said 'this'll take some shifting'. At which point I, rather taken aback, said, 'it ain't going anywhere, but do feel free to replace your fence on your property'.
After an attempt from me to smooth things over face-to-face they then sent me a note reiterating what they wanted and then refused to talk to me - basically because they weren't getting their own way and were mistaken about the boundary. I think they probably wanted me to contribute to the cost of the fence too, but we never got that far. They thought their fencing contractor was an expert on boundaries - mistake: I worked for a property developer at the time and had advice from our in-house surveyor and a lawyer. NDN came round and threatened me on my own doorstep. He'd been drinking, which added to the experience 
. I didn't let him intimidate me. In the end, they replaced their fence and everything stayed as it was.