On the boundary line between my back garden and my next door neighbour is this tall tree, situation looks like attached pic. But the trunk begins to grow from the neighbour's side but leans heavily into my side. Most of the tree is on my side of the boundary. 

My issue is that it blocks most sunlight hitting my small garden and through my windows, it sheds leaves everywhere in autumn, it looks horrible and blocks my view from my windows. Also it sways heavily in stormy weather which always worries me. I want the tree trimmed down, I hate it. 

The thing is, nobody has lived in that house next door for at least 5 years, possibly more. I have no contact information for anyone who owns the house. Nobody visits or stays there. I don't know their names.

Legally, it's a shared tree so I can't just cut it down without permission. I know a tree surgeon who told me not to touch it or I'm in for a serious lawsuit once the owners find out even though it's mostly on my side. Potentially could cost me tens of thousands he told me for criminal damage. 

I know there is no "right to light" and I know I am allowed trim anything overhanging my side, but I'm also not allowed to damage or kill it or trim it too much that it causes any damage, so trimming a little bit of overhang won't help my issues at all. It's the height that's the issue. 

Is there anything I can do without having to try chase down some owner and blindly phone them asking for permission to trim it? 

Or how serious of a lawsuit am I looking at if I just topped it? Not even kill it, just trim the height. 

I've been living here about 2 years now in the hope somebody moves in so I can ask them for permission but do I really just need to wait forever?

Dealing with a shared tree on the boundary line