What are my rights if my neighbours tree encroaches onto my property?

Under common law
You may prune unprotected trees or hedges overhanging your land without the consent of the owner but you must not trespass on to your neighbour's land to do this. In addition you must offer to return the timber and any fruit attached or which has fallen on to your land.

It is also considered neighbourly to notify your neighbour of your intentions prior to carrying out the works.

The above may also be applied to root encroachment.

Warning
you may become liable for criminal damage if such work causes the tree to die or become dangerous.

Recommendation
It is recommended to carry any tree works in accordance with good horticultural or arboricultural practice. Poor tree work can be counterproductive and is often potentially dangerous.