I'm sure VHG is quite right about the legalities Dilly. This is quite a common problem.
The neighbours may have a right to have their tree there, but no right at all to cause damage to your garage. You have every right in Law to complain to them, and have them recompense you. The trouble is of course that the last thing anyone needs is an ongoing battle with the neighbours.
As a builder, if I was asked to deal with it, I'd certainly ask to take a look, with them, from their side, to explain the problem.
Builders are thick-skinned by necessity. Some of us can be quite diplomatic. Find yourself a local builder/handyman who would be prepared to talk it over with them on your behalf.