There are a number of issues here and an awful lot of posters who don't know the difference between being the registered keeper of a car and the owner of it. It's not the same thing people.
Firstly, as others have said, she was clearly "fronting" and that's fraud and the insurance is invalid. That however isn't terribly relevant here as the car had been sold and neither mother or son was driving it or owned it at the time. They cannot be held responsible for an accident in a car they no longer own.
I'm assuming son hasn't got a new car? There's no suggestion from the OP in either post that he does despite several answers presuming it.
If the car is on HP then he doesn't own the car in the first place and cannot sell it. I suspect this is a case of someone finding they cannot afford to run a car and getting shot of it unofficially for a sum whilst continuing to pay the finance off and insure it. In which case there's all sorts of problems to sort out. Not going to be easy to sort this one out I suspect, but in answer to the original question, they still won't be responsible for somebody else's accident. If the other driver had no insurance of his own then it will be treated as a driver uninsured case.