Hi BML
I work as an arboricultural consultant and deal with similar situations on a daily basis. Looking through the thread there are some truths however there are also some grey areas. Under common law your neighbour has the right to prune back branches (and roots) to the boundary but can not enter you land and must offer all the arisings back (otherwise it is theft), unless you live in a Conservation area or the tree is subject to a Tree Preservation Order (TPO). In this situation a notification/ application must me made to the Local Planning Authority stipulating the works. From what you are suggesting it would be likely that the Tree Officer would refuse on the grounds that the works don't comply with BS3998:2010 Recommendations for tree work. Speaking to your local Tree officer may help and they may think a TPO is appropriate given the immediate threat however be warned the local authorities can't just TPO any tree it must be expedient in the interest of local amenity i.e. if you live in the middle of nowhere and the tree is not of public benefit they may choose not to get involved. Good luck.
Regarding pruning, avoid spring and late summer/autumn. With most broadleaves the best time to prune is a month or so after buds burst in the spring. Bud burst and leaf senescence uses a lot of energy, give the tree a bit of time to photosynthesis before damaging it.