What a lovely daughter you are! You sound so sensible and loving and your gut instincts are absolutely spot on.
Your mum, whatever her problems, did a brilliant job in raising you.
I can understand both sides of this. I can empathise with how she feels and I believe that your mum's anxiety will ease once the baby is born and everything settles down but before that you have that difficult time to negotiate when the baby is new.
My son has two boys and they live quite a long way from me. I have tried and tried to be more involved in their lives but my daughter-in-law has a hands-off mum and finds it difficult to accept. The result it that if her own mum isn't involved, she doesn't want me to be.
I have learned to live with this and see them when I can.
It was a very different kettle of fish for me when my daughter announced she was pregnant. She too lives a good distance away and I went through a very anxious time feeling that this was my last chance to be a hands-on granny, not helped by the fact that son-in-law's mother lives nearby and was going to look after the baby two days a week when my daughter returned to work.
I was envious and quite sad that I couldn't see them more. Quite frankly, I still am but have accepted the situation for what it is. I feel that the most important thing is that my daughter has a happy marriage and loves her little boy to bits.
As it worked out, mother-in-law found that she was being over-ambitious having the baby two days a week and soon cut it down to one (causing all sorts of problems in the meantime!).
I suppose what I'm saying is, try to keep your mum as calm as possible - personally I think that a schedule for visits is a perfect way to allay her anxieties. If she knits, crochets or sews, give her projects that will keep her busy and make her feel valued (but make sure she sees you using them afterwards!) Eventually when the first few months of chaos are calming down, I think she will settle down too.
I accepted that my daughter didn't want me to go to stay with them when the baby was new - she wanted it to be just her and her husband - but I went to stay for a few days when her husband went back to work. I mostly stood guard over the little one while my daughter got some much-needed sleep.
Try to relax, even if your mum can't - I think all will be well 