With a new baby coming, you might consider hiring a college student to do a two hour shift starting at that time. That way, the college student could pick up your preschooler, drive him home, and then help grandma with the baby/housework for a little while and give grandma a break. If the baby is difficult more fussy, then grandma might enjoy having time to focus on older grandchild exclusively at that time every day. But even if the baby is "easy" grandma will likely appreciate having someone else to fold laundry, clean up messes, wash bottles, and so on for a little while when she is juggling two kids. $250 a week would be pretty tempting to a lot of college students.
You could actually look at high school students to, since your child could be walked home in a stroller. It sounds like it is just your mom's health that makes the distance on walkable. If the high school in your area gets out before then, and if there are any students who are not constantly in afterschool programs, this would be a great job for a highschooler. Walk to the preschool with the stroller, walk the child home, help grandma for a little while, then walk back to their own house.