What should I do with the trimmed branches?
Ironically, even though the branches belong to your neighbour, you cannot simply throw them back over his fence. That could be deemed to be fly tipping of garden waste. Advise your neighbour that you intend to burn them or take them to a recycling centre.
How far can I trim the branches back?
You can cut branches and/or roots back up to the limit of your property, your garden fence, say. There you should stop.
You are not allowed to go into a neighbour’s garden without permission to cut a tree back. Nor can you lean over into his garden to cut back the ‘offending’ branch – you will be trespassing. If you cut the branch back beyond the limit of your property, into the very trunk itself, say, you could be liable for damage or trespass.