The son is already 100% liable for rent – that’s how a joint tenancy works, the tenants are jointly and severally liable for rent. The landlord can pursue either or both of them. If one isn’t paying, you can be sure they’ll go after the other. If there are arrears because one person hasn’t paid their share, they can both get evicted.
Before 2/2/22, this isn’t the HA’s or the son’s problem though: you have an overpayment recoverable from the mother only for a few months before that (and good luck with it – no ongoing HB to recover from and only care home pocket money as income). From 2/2/22 onwards, full rent is payable by either or both of them, how they share it out between them is up to them. Realistically the son will probably have to pay.
You might be able to find a way of reducing the overpayment in line with the mother’s changing realistic intentions: perhaps he only reported the change when he was finally sure that she would not be returning.
