The OP has said repeatedly that she feels bullied by her mother.
And she was raised by her, so I'm going with that rather than what you think.
It's easy to see why, if you were brought up by the kind of woman who would throw a toddler tantrum over your of childcare, that you might be vulnerable to emotional blackmail and bullying.
"There is nothing actually wrong in the gran wanting to know what the arrangements will be when the child starts school - it does affect her life too."
The mother doesn't even know which school he will be going to.
The people whose lives will be most affected by this are the little boy and his parents.
When they have had a chance to figure out arrangements, then she can ask how it affects her. To make a fuss about knowing so early is just acting like a spoilt little baby.
It affects her life too 
She's a grown woman who will see her grandson on a different schedule. Normal people do not need four months notice of that kind of minor change.
The level of pressure she is bringing to bear about sorting out her grandson's life to suit her would make me worried about how she is going to respond to other changes as he grows up.
I'm not sure someone who wants to control a 4 year old's weekly schedule for her own enjoyment to that extent is a healthy influence on a growing child and their burgeoning independence.
You can just imagine the shit fit she'll throw if the child arranges a play date on "her" day.