In short yes and the wood cut should be returned to land owner. However the amount m you will need to cut back will have detrimental effects on the tree and this now becomes a grey area. Who is the land owner neighbouring your? I would contact the council and ask about tpo and advise on how to proceed

Yes you are legally right to cut back any over hanging branches back to your Boundry lines

Yes .you are in your legal right cut back any over hang to your boundaries, as long as you offer wood cut back to owner,not normally many takers!!!l
Different situation if trees have preservation orders

If the tree is TPO you cant do any work to it without permission from the council. If it is conservation area, you need to give 6 weeks notice of any works. Otherwise you can cut any tree that overhangs from your neighbours garden back to the boundary. The law says you should 'offer' the cuttings back to the owner. But in reality few people would want them back.

The wall is buckling.

This is now a direct structural damage investigation that requires an consultative report opinion, before any tree works commence as a civil action may follow.

Hello, firstly I'd speak to the local council to see if there is a TPO, as you may have to apply to cut it. Without actually seeing the tree it would be impossible to say if you'd damage the tree, but a qualified professional would be able to advise when they came to quote. As for where you legally stand you have the right to cut it back to the boundary line but they have the rights to the wood, if they want it. Hope that this has helped.