My advice would be: Write to the council, recorded delivery, making it quite clear that, having notified them of the damage being caused by the tree, you will now hold the council responsible for any structural damage to your house.
Moreover, if they do not take appropriate action within 30 days you will also hold them responsible for
(a) the cost of any experts* you deem necessary to evaluate the situation or recommend options; and
(b) the cost of any legal action necessary to compel the council to act.

* ie a tree surgeon to say definitely that the damage is being caused by a tree and that it needs to be cut down or have its roots cut back or whatever


However - rshdunlop makes a very good point: discuss this with your insurers now. They may say they'll sort it out with the council (in which case you can ignore my advice above) or they may shrug their shoulders until you actually need to make a claim (in which case go to Plan B above and write to the council).