I'm not sure the Council will take action if the tree is on private property. I'm pretty sure it will be the responsibility of the house owner to take the appropriate action and I think you should write formally to the owner, sending the letter by registered mail so you have proof of delivery, expressing your concern about the danger of it falling and injuring somebody and also pointing out that it has damaged your patio. Ask politely to have the tree felled, the wall made safe and the damage to your patio made good. Also state a reasonable date by which you expect the action to have been taken. However, I think it would be polite to talk to your neighbour face to face first and tell them that the letter is coming because you are taking this issue various seriously in case somebody gets injured if the wall falls.
I would imagine that the owner will have some third party liability cover as part of their household insurance which will meet the cost of the claim. However, if the owner fails to provide a satisfactory response, then I think you should consult a solicitor.