In reply to j0ntyg:
As DaveyGardner has already posted, ANY TREE of diameter within a Conservation Area is subject to protection (provided it exceeds 75mm at 1.50m height and is not a fruit tree). To do work on the tree you apply to your local planning authority, using a form normally available on their website and describing what work you wish to undertake (in your case, branch removal at your boundary line); they then have 6 weeks to consider your application. They can then either put a Tree Preservation Order on the tree, or leave you to get on with the work.
Assuming that they are not bothered enough to slap a TPO on the tree, once the 6 weeks have elapsed you can then do the work. In terms of land ownership law, you can remove branches that overhang your property back to the property boundary but no further. In theory the cut branches do not belong to you and you should offer them back to the owner on whose land the tree is growing - although in practice a municipal landowner would not wish to have them back.
In any dealings with trees of any size in a settlement and particularly a Conservation Area, be aware that many people will hold a view of this tree that is opposite to your view. Trees add significantly to character, wildlife value and property values. So, any precipitate action you might take would be likely to irritate a lot of people and lead you into hot water (it is an offence to wilfully damage or destroy a tree in a conservation area without having first notified the planning authority). I would not recommend the use of a drill and herbicide - it is not your tree to destroy and so you would be committing criminal damage as well as an offence under the Conservation Area legislation.
From what you say it is a problem you have had for a while now, and so a 6 week rocess should not cause you too much concern.
Lastly, once you reach the point where you have consent and are to do the work, consider using a member of the Arboricultural Association to do the work properly and with the right insurances in place. Any old fool can buy a chainsaw and then drop a branch on something they shouldn't (or prune the tree in such a way that it makes your situation worse in the longer term).
Excuse the lecture. It's just the facts!
Nick