YABU to want to hack into the tree. I know it's a bit annoying, as it overhangs your land, but it would look shit chopped in half.
Why not just leave it? It looks nice. It will look sweet with Christmas lights on. We have a boundary with our (lovely widowed) neighbour, and a fir tree she planted a few years ago has tripled in size, (gone from 3 feet high to 9 feet high,) and a couple of feet of its width comes over onto 'our land.' She actually asked us if we want her to chop it, and we said 'no it's fine.'
We don't mind at all, and moved part of the 3 foot high wire fence to accommodate it's growth (it's at the side, near the front, about 45-50 feet from our house, but it's still overhangs our garden very slightly.) We don't mind though as it's a gorgeous tree. And she is a lovely neighbour.
Just leave it. I would. I mean, when your neighbour planted it, there was always the risk that a new neighbour may not want it hanging over their garden, and maybe you are entitled to feel irked. But why fall out with neighbours? I have had God-awful neighbours in the past and it has been a living hell and has caused anxiety and depression.
But where I have lived for the last 7 years, all 10 sets of neighbours in my little part of the road are amazing. Lovely kind people, ranging from couples and widows aged 45 to 80, one family with 3 kids, one family with 2 kids, and a 19 year old couple. Everyone gets on so well, and people just don't complain; if something is a bit 'awkward,' they work around it.
Like the old lady that lives opposite occasionally has a visitor who parks so I can't swing into my drive, but I park a few feet away (on the road) and then move the car back when they've gone. It happens about twice a month, and I don't care.
It's the best neighbourhood I have ever lived in. We are all good friends, we help each other out in an emergency, we give each other plants and strawberries and tomatoes etc, and we do favours occasionally if someone needs help. Some summer days, there are 10-12 people or more in the close chatting for half an hour. It's lovely.
I would never sacrifice that because of a frankly quite pleasant tree over hanging my garden.