Sorry that sounded a bit snippy.
And also is what I know about England and Wales.
Here's a fuller explanation:
You have a common law right to cut back any branches (and roots) from your neighbour's tree that come onto your property (above or below ground) up to the boundary line.
You don't need permission to do this but you must do so from your own land.
Also, you must not cause the tree to become unstable/unhealthy by pruning. If you cause the tree to die or become dangerous, you could be liable for damages.
The cut branches still belong to your neighbour, (the owner of the tree) so you have a legal obligation to offer the cuttings back to them and if they do want them, best practice is to actually return the branches to their property to avoid accusations of fly-tipping. Don't just throw the branches back over the fence - this could be considered fly-tipping or "nuisance".
(It was certainly a nuisance when it happened to us - and it wasn't even our tree!)
If they don't want the branches, then the responsibility for disposing of them falls to you.
Before you do anything check for Preservation Orders- If the tree is protected by a TPO or is in a conservation area, you will probably need permission from your local council before carrying out any work, even cutting back overhanging branches.
Hope this is more helpful.