Aside from whether she's potentially putting herself at risk for some kind of fraud or breach of contract by taking out the finance for someone else (idk if it actually is, guess it depends on the small print...), she's also going to be on the hook if he stops paying, or damages the car and his insurance value it less than the finance company etc.
Also why on earth is he looking at a £35k car if he's so skint that he's got persistent debt problems? I'm not blaming her, but she is enabling some very poor decision making on his part. I know car prices are still a lot more than they were 5 years ago, but he could spend £10k on something that will still be spacious and comfortable and reasonably reliable, and have a whole lot more money left over for paying debt down or saving.
If he really cares so little for her and his kids that he's willing to blackmail her and cut her off if she doesn't facilitate this then that's on him, not her. Sometimes standing up for yourself is the right thing to do.
