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As an update to this, I'm pleased to say this has been sorted out. With some difficulty. The finance company (it was a HP agreement) wouldn't do anything until they had heard from the car dealer, the car dealer wouldn't engage with the finance company until they had assessed the car.
The car was not safe to drive, and they were insisting that she had the car returned to the dealer. After much to-ing and fro-ing, and some strongly worded e-mail as to her rights under the law, finally the dealer agreed to collect the car. A couple of weeks after they had done so, the finance company contacted her to say that the finance agreement had been cancelled.
Moral of the story is, if you are going to buy a used car, which is fraught with possible problems, buy it on finance on HP agreement. The interest rate may be higher, so in the long run you pay more, but you do at least have their support as the car is legally theirs, so they have an interest in it. Luckily this took place in the first 30 days, so she was allowed to reject it. Had she had it for longer, it would have been more complicated.