If it was just the texts in the beginning then maybe I would send over a couple of pictures once baby was born.
However. Threatening that there is nothing your dd can do to stop her seeing the child? Nah. The woman should be grovelling instead of threatening to come to the hospital. She doesn't give a shit about your dd or the grandchild, it's clearly all about her and her perceived rights. She hasn't asked about how either of them are doing. It's all me, me, me.
This person has got NO rights to see the grandchild because there aren't automatic grandparent rights. If she had a proven relationship with the child and the dd stopped contact, then the gp could apply to the court to ask for permission to apply for contact.
At the moment with the dad, unless he is present he cannot be put onto the BC. However, this does not mean in a week, month, year, whatever down the line he suddenly gets a conscience he can apply for rights and contact through the court.
There's plenty of advice about this on government websites, family lawyers websites, gingerbread etc. If that's not enough there's also places like cab who will advise her the same thing. Gp's have no automatic rights, and unmarried dads have to be there at registration to be put on BC, and if they want access or rights if not on BC he applies to the court.