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Although you may still need permission from the council if there's a TPO on the tree of you're in a conservation area.
https://www.gov.uk/how-to-resolve-neighbour-disputes/high-hedges-trees-and-boundaries
Also doing it without asking first can easily lead to problems between neighbours (this happened to our neighbours and their neighbours on the other side).
We had a holly tree overhanging our garden and we agreed to pay half of the cost of getting it chopped down (with council permission as we're in a conservation area) as I hated the prickly leaves it shat all over the grass/beds in our garden.
Yeah, they are entitled to trim whatever sits over their boundary.
Sounds like they might want it gone though?