You are allowed to cut any part of your neighbours tree that encroaches onto yoru property. The material you cut belongs to the owner of the tree and you should give it back to him.
Have you tried speaking to the neighbour and explaining your problem? This is by far the best way forward.

the previous answer is correct, however, if you attempt to root prune in any way and the tree suffers any failure, be it disease or collapse of the tree, you could be held responsible and legal proceedings could happen! and technically, you only have to return prunings of anything bearing fruit to a neighbour if you cut it!
talking to the neighbour is definately the best first action, then maybe obtain a written report from a recognised and insured arboriculturist.

Many thanks. I will, of course, speak to my neighbour - who is a reasonable individual. I have the feeling, however, that any remedy might be expensive and I think he would not want to take these trees down.

It seemed wise to check the legal position before speaking to him - not with the intention of starting off by quoting the law, but just so that I know ultimately what our respective responsibilities happen to be.

You could ask at the local council if they have an arborist who could explain the legal situation to you.

On Escape to the country on BBC 1 last night, a high Leylandi hedge near the house in a quite large garden may have put clients off buying, for between 550-600K, a house that had been on the market for some time. The Leylandi got several mentions from the man, who was a keen gardener, who might have bought, had these pernicious trees not been a potential source of bother. Check it out on iplayer. Slide it in as another reason why your neighbour might agree to reduce height or remove.

I think that the law has changed and that you can get them cut down but YOU NEED TO GO TO THE COUNCIL to get this right.