My opinion is that the damage was caused by neglect and the insurance company would wash their hands of it.
I would formally ask them in writing, the letter sent by recorded delivery, if they have already said they are going to do the the work, to accept responsibility for the damage caused. If they refuse to do that or give you a date when they will come and do the work - tell them - again by letter that the work will be done by another company and the bill will be forwarded to them and a claim made in the small claims court.