He will not engage with social services
This will not stop them assessing the situation. Or kicking him out if they find he has no legal rights to live there.
Your mum refused you POA. She refused to allow you legal responsibility to sort it out. So the solution is simple - wash your hands of it. It isn't your responsibility, in any way, to sort this out. Your mum did you a favour TBH.
So just stop worrying about it, it's literally nothing to do with you whether the place is sold or not, whether brother is kicked out or not, whether anyone pays utility bills and council tax or not.
Utility/council debts will be your mum's, not yours, and if she's in a care home with no income it doesn't matter if they take her to court and win even, they're still not getting paid. Her debts or CCJ's don't matter, she's not going to be applying for credit any time soon.
Don't even bother telling the utilities etc your mum's in a care home, you don't have to give them that information, it's nothing to do with you. If they send debt letters ignore them, they're your mum's letters, not yours. New owners can tell them she no longer lives there, it won't impact on them either because it's not their debts. If utilities are cut off that's your brother's issue to sort out if he's still living there or new owners issue to sort out when they move in.