Hi,
If the tree falls and damages their property it could be a blessing for you. It is a costly process civil procedures to resolve some matters which may seem normal.
Assuming you have evidence of tree damage on your property for example entering waste drains, effecting the structural foundations of your building, garden or foot path & fallen branchs
Endangered species, most councils would not approve of the removal unless you can prove the above.
Keep copy of all communication to there OC. Arranging discussion is one thing, how they react is another.
Good Luck.
Digby