Yep, I would contact you council as PP's have suggested.

The damage done to your spa etc. can be claimed on your home and contents insurance. They might also have advice about the trees?

I *think* (but I'm not certain!) that although the tree is your neighbour's responsibility as it is on her land, the branches which overhang your fence are your responsibility?? Maybe someone else can clarify?? (I'm sure the council can!)