I think your queries are best directed to the council as they would have dealt with onerous neighbors like this all the time. They should advice decisively.
The branches do need to be returned back to them. I would strongly advise you not to dispose of them yourself (throw or sell or give away) - it might be technically theft or handling stolen goods, all depends on the jobsworth who picks up the complaint.
When you drop it off over the fence, make sure it doesn't drop on their beloved petunias or whatever, it could end up getting leveraged as criminal damage.
The branches do need to be returned back to them. I would strongly advise you not to dispose of them yourself (throw or sell or give away) - it might be technically theft or handling stolen goods, all depends on the jobsworth who picks up the complaint.
When you drop it off over the fence, make sure it doesn't drop on their beloved petunias or whatever, it could end up getting leveraged as criminal damage.