Some great, sensible advice here. I'm a fairly new GM, and I do confess that I too felt a little anxious about how often I would see my GC, especially as we live a very long way from my DS and DiL. It didn't help constantly seeing photos of my friends with their GC, who all live very close to each other, and are very much 'hands-on' GPs. I did feel that I would be missing out. However, as the months have passed, they do visit us to stay for a few days whenever they can, and we visit them for the odd long weekend. Facetime & Skype is brilliant, and we chat a couple of times each week. Things are working out just fine now. In fact, when they leave, my husband and I breathe a little sigh of relief to have our own peace and quiet again!Lol
From my own personal experience, when I first became a mother, we did get into a regular routine of seeing GPs at weekends etc., only to find that it really restricted our own lives, when occasionally we wanted to do something different, on our own, and we began to feel trapped in a situation that we didn't know how to get out of without upsetting our parents! So, from that perspective, I would suggest not to make any definite plans about exactly when you see your mum, or have her come to stay with you at weekends ... it will be very difficult to get of, without upsetting her. Far best to have a casual arrangement, and arrange visits when it best suits you both.
Becoming a parent is such a special time, but it is also very special for GPs too. I think they way she is feeling is quite common, but it needs to be nipped in the bud before it escalates. I'd have a gentle little talk with your mum, and let her know that there is no need for her to worry about how often she gets to see the baby. Let her know that she will be an important part of her GCs life, and that her worrying about it is upsetting you when there's absolutely no need. And let her know that if/when you need advise/help, before/after baby is born, you'll be giving her a call.