I don't understand why your mum needs the car - in particular - to get her to her job ? Maybe I've missed something, but why can't she use the van for that ? Seems as if the only reason she wants the car is so she doesn't have to pay fuel costs.
Now, yes, she is helping you out by collecting DS after school, but is that contribution worth £60 a week ? What would it cost to have a childminder do the same thing ?
Playing devil's advocate, maybe, just maybe your mum had begun to feel resentful at picking DS up (don't know how long she's been helping you for ?) even if she was happy to do so previously and this is her cack-handed way of expressing that resentment ? Quite obviously though, she knows exactly what she's doing by her blackmail-esque remark ..... which I find very hard-faced indeed (instead of talking to you about the issue if she felt she was being taken advantage of).
I really would be tempted to resolve the whole issue by getting a childminder - if the cost is less than £60 a week, what have you got to lose ? I know that's easy for me to say, because I do appreciate how easily this might blow up into a family feud, but if she's blatantly using DS as a means of getting a considerable amount of free petrol then I'd be concerned about what she might threaten next (do this for me, do that for me, or I won't pick up DS). Okay ..... hopefully it would never come to that, but you really don't want to be beholden to anyone regarding childcare. Pay someone to do it and you're not.