Your mother owned the house, but the council have a claim on it for that part of the mortgage that has not been repaid.
When she died, there seem to have been significant debts and costs: the unpaid portion of the mortgage; the nursing home costs; and the funeral expenses.
The assets available to meet those costs seem to be a relatively small amount of savings, and the house.
Things need to be tidied up:
1. Money has to be found to pay the nursing home bill, and I don't think there is any scheme under which a claim can be made: the Fair Deal scheme is the government support, and repayment is part of the deal.
2. You can not simply allow your uncle occupy the house and pay the mortgage. Things have to be regularised. If she did not make a will, the house is in effect the property of all her children, and that has to be sorted out. Do the family wish to hand their interest over to your uncle? That involves legal work, and it's not a DIY job. Then the question arises: if he gets the house, do the family want him to accept responsibility for the debts? That looks to me like the agenda for a family conference. The council might agree to the mortgage being transferred to him, but somebody needs to make contact with them to discuss things.