Suffice to say, my firm view here is that the tree blocking a view is a legal “red herring” in terms of justifying it being chopped down.
Another important point is that the tree trunk is on your land. It is therefore your tree. It is your responsibility and by definition should be your decision to fell it. Where branches overhang your neighbours’ land, they may trim those branches, but should not interfere with any part of the tree on your side.
A caveat to your freedom to do as you wish with a tree on your land, and also your neighbours’ right to trim overhanging branches, is that permission is always required for work to a tree that is subject to a Tree Preservation Order, or that is situated in a conservation area.
TPOs are quite a well-known concept, but clients of mine are often surprised to learn that if they live in a conservation area, all trees are deemed to have a TPO. This means that anyone wishing to work on such a tree must make a formal application to their local authority.
You do not say whether you live in a conservation area, but if you do not, I would urge you to contact your local tree officer or if there is no such person, the local planning department to see if your tree can have a TPO placed on it. Local authorities will often issue a tree with TPO status if they fear it is at risk. If this can be done without you being implicated, it may also ensure that all important cordial relations with your neighbours are maintained as well.