The general rule
In England you can usually cut back branches and roots that cross onto your property, up to the boundary line, without your neighbour's permission. You can't go onto their land or cut beyond the boundary without consent.
This is general information, not legal advice.
Check it isn't protected
If the tree has a tree preservation order or is in a conservation area, you need council consent or notice first, even for branches over your side.
The cuttings
Technically the cut branches belong to the tree's owner, so it's polite to offer them back. In practice most people agree for the contractor to remove them.
Talk first
A friendly conversation usually goes a long way. Often neighbours are happy to share the cost of having the whole tree properly reduced, which looks far better than a flat cut along the fence line.
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