Your neighbour's extension is causing a legal nuisance, both with the overhang and the discharge. If a friendly discussion goes nowhere then write to the neighbour stating your concerns and ask for it to be rectified within a reasonable timeframe, or you will arrange for the work to be carried out and sue them for the costs. Ask on
http://www.gardenlaw.co.uk for more assistance though because there are some circumstances where you may not be able to do anything (such as if you didn't object at the time it was constructed, but could reasonably foresee the rainwater would have to discharge onto your land).
Beware - some of the advice given in this thread is inaccurate.
The direction the good side of a fence faces bears no legal indication as to who owns it. It never has done. Although widely reported as fact by people who should know better, it means absolutely nothing.
Although the law presumes that you will fence the furthest extents of your land (which with a close boarded or wire fence would suggest that the post is normally sat on your land with the facing on the boundary) there is no legal requirement to fence your land and you could site a fence 2 metres inside your property if you wanted. If you didn't maintain the land on the other side you would probably lose it over time if someone else did, but if you did or there were other markers in place then any court will find the land is still yours.
Even if there are ownership marks on your deed/title that suggest the boundary fence is owned by you, that is only the case if it is the original fence or a new fence was erected in the same location (regardless of the party who paid for it). If the current fence was erected on the neighbour's land and then the original fence was removed, it is no longer your fence.
Additionally, be aware that boundaries are defined on the ground not on paper. The measurements in your deeds or any LR titles will not be accurate and courts will treat physical indications with a much higher priority than written ones.