If the tree is not on your property you are able only to cut down any branches that overhang your property. Google "High Hedge Law" and see if this applies. However if there is only one tree then this law will probably not apply.

Is the tree in question dangerous? If so there is the Dangerous Tree Act. Speak to the horticultural officer of the Council for advice?

The best solution is to appeal to your neighbour who owns the tree. It is really for him to decide what to do. He might not want to go to the expense of around £1500 but you could offer to pay or share the cost.