http://www.gardenlaw.co.uk/trees.html is quite a useful website. Can I caution you to be a bit careful about tree removal as trees draw water from the soil and keep the water table in balance. Removing a mature tree, especially on clay soil can cause the water table level to become unstable and result in damage to foundations.
Locally, the only house on our estate that has subsidence problems is the one where a large tree has been removed. Other house where the gardens who have kept their trees are okay (touching wood here!)
...It may be worth getting some expert advice before talking to your neighbour. Root pruning and containment can also be an option.