Neighbours tree close to house causing issues with insurance

[Zippee]

Our neighbours have a silver birch tree in the back corner of their garden which is approx 5 metres from the back corner of our house, as such we're finding a few insurance companies are ramping up the buildings cover price - especially as it's the height of our house. Is there anything we can do about this tree? I know we can remove any branches overhanging our side, but to the left theres one branch about 1/2' thick thats within a metre of our house but on the neighbours side, plus the top branches block all light to our garden in the evenings. Obviously we will approach them and ask if they'll mind a tree surgeon trimming (we'd offer to pay) but in case they say no can anything be done due to proximity to our house? Would they be liable should the roots cause damage or it blows down? Also how far do the roots on these trees burrow?
Pis attached showing the tree (already pushed out a concrete fence post/panel) and a frozen wife! The trunk is right in that corner of the fence.

[Zippee]

RemainAllHoof said:

One issue is that even if you trim the branches, the roots will extend under your house and bugger it up. I suspect that is why they are increasing premiums.

Exactly my worry. As such can we force removal of the tree at all? Or ensure the neighbours are liable for any repairs as a result?