I've got a beautiful tree growing in the bottom corner of my garden that I love ,although it has grown to around 20ft which has caused a slight issue with one of our neighbours . I do thin it out regularly to allow as much light through as possible but I'm not cutting it down . I've said to our neighbour that he is welcome to cut the branches overhanging his side but its not coming down . I can feel a Swampy moment coming on .
I bought a plot in Dodworth 2 years ago where I now live, with 2 large mature trees on it. A couple were interested in the plot next door, which also had 2 mature trees on it. They approached me and said they were going to apply for permission to chop down the 2 trees on their plot, and they said I would also have to chop mine down as they would block the sun in their garden. I told them I would oppose their application (the trees have a TPO on them), mine were staying up, and I was going to plant some more trees anyway. They didn't buy the plot. I love trees.
If the tree is on the boundary line consult your title deeds to see who owns/has responsibility for the boundary and maintaining boundary.