Based on your last post I just maybe add that if it was lopped badly previous and she doesn't like it now, then perhaps agree with her and say that your DP would like to discuss it further with her as he's confident it can be properly re-shaped to grow in such a way that that would address everyone's concerns (her privacy, shape of the tree, your daylight).
As for some posts on here - goodness me, how miserable.
The OP knows she should have investigated further but it's done now.
Equally she's proposing managing the tree, not cutting it down.
My NDN's have a huge tree that's adjacent to our garden and every few years it needs a tree surgeon to shape/lop it not just for our mutual benefit (light and branches over the driveway) but for H&S reasons (in a storm a huge branch fell and could have caused serious injury, but luckily it was only a car that suffered).
We chip in and pay 50/50 and seem able to speak civilly with NDN's about what needs to be done - obviously being guided by the tree surgeon. I started with a letter to NDN's hence the suggestion.
It's a bit of a PITA as it has a TPO so we need permission for works, but it now (7 years since we started engaging with the TS) looks stunning for being properly maintained and something we can enjoy rather curse because it's a nuisance/danger.
Happy neighbours, happy tree....win win.