Sorry about your loss, Steve. I know it must be hard for you. I had to deal with these kinds of things recently myself. Since your location is listed as in the UK, my guess is that your case will be similar. The banks are legally bound to pay funeral costs out of the frozen bank accounts, provided that there's enough money there to start with. In our case, we just handed the invoice from the funeral in to the bank, and they paid it directly.
Also, they have to pay out any money that is clearly designated for your mother - i.e. if there are any wages or anything like that that's in her name, they have to free that if she can prove that it's hers. And if there is no will, depending on how much is left, they should transfer the bank accounts into her name, but from all I understand, that is down to how much money is involved. In our case, there wasn't even any probate, because there wasn't such a great lot in the accounts to start with.
I hope you get it all sorted soon, sending some vibes of patience and strength your way. I know how much of both is needed in times like these.