You do not take another person's five year old anywhere other than where you have been given permission to go. If this means picking up your own kids at school late then so be it. You just simy do not do it.
We don't know how old the other children are. If they are 6 & 7, and too young to be left unsupervised for 30 minutes while the mother takes back the kids from the field trip, then the other mother may not have had much of a choice.
If her children are at school they are not unsupervised so that doesn't matter. And besides that is her problem, no the OP and not the school even though the activity ran a little late. It doesn't excuse taking someone else's child somewhere you have no permission to take them.
You are right, it was the drivers problem and she solved it. I don't blame her for putting her childrens' welfare above another parents.
She solved her problem by breaking the rules! Her children's welfare was not in danger--they were at school.
Her primary responsibility upon volunteering to drive is to carry out the responsibilities given to her by the school. If she cannot handle that and her own responsibilities as a mother, then she shouldn't volunteer.