Just to add that I don't know what the current regulations are, but at one point someone from the council went round to my (late) parents' home to check that they were getting all that they were entitled to.
We hadn't applied for Pension Credit because they had savings. However it turned out that, because they each received Attendance Allowance, they were deemed to be one another's carers and this meant that they had an underlying entitlement to Pensioner's Credit. It was stopped when Dad died.
I'd suggest that you get in touch with Citizens' Advice - they helped a neighbour's friend apply for various allowances.
In the end, Mum's care was supplied by a company subcontracted to the council and we paid the company additional money from Mum's AA to supplement the care.
The care company gets a massive amount of money out of it, though, and it's certainly not passed to the carers.
One time, my husband had gone up to the care company's office to pay the private bill. He handed over the cash that I'd drawn from Mum's account to pay the bill.
They later denied that we'd paid and demanded the money all over again. I told them we had a receipt. They demanded that DH take the receipt to their office...
It was sorted out, but I never trusted them again - with good reason. It was a franchise, I believe, so other companies of the same name may be more trustworthy.
Carers who had left the firm used to refer to it as "Scarewatch" - it rhymed with the actual name.
The firm that we used afterwards - again in conjunction with the council - was Avenue. They were better.