You really need to start managing this or unfortunately you're likely to fall out once the baby is here, especially as she lives close. Wanting to hear about your appointments?! No.
You may feel it's like kicking a kitten now, but when you've got a 3 day old you're trying to learn to bf with, you've had no sleep and she is harassing, trying to get between you and the baby and you building your sleep/feed/bond relationship, you will want to take that kitten by the scruff of the neck and drown it without a second thought. And if she's generally nice and you get on, that will be a terrible outcome!
For HER sake too, sit her down with you and your DH - or get your DH to do it. Tell her that she is beginning to go too far and stressing you out before the baby is even here. Tell her that you feel so anxious about her doorstepping at the hospital that you've decided now in advance that there will be no visitors at the hospital at all, you're so worried about the stress and pressure impacting on your labour. (You can always relax this later, but put an end to it all by telling her she won't be told when you're in labour and if she comes to the hospital she won't be allowed in).
Remind her that your appointments are about YOU, your personal medical affairs - and ask her to stop asking about them.
Above all, remind her that when the baby comes, although she'll be seeing lots of you, you will want to be spending the majority of time alone, with baby, bonding and feeding and sleeping, and that in order to get bf going that the baby will need to be with you, and mainly you.
It may seem heavy handed, but it's far preferable to a screaming stand up row in the first couple of weeks. If she understands NOW that no, she won't be spending all day every day at your house cuddling the baby, that's a good thing - and you can tell her that too, that you KNOW it's going to be tough for the first few weeks and you know how enthusiastic she is and that it worries you that she will try and take over because you most of all don't want to have a falling out.
I find that these 'it's like kicking a kitten' people tend to be very good at the kicked kitten eyes to get their own way. Bottom line is that someone with good boundaries who cared about how you feel wouldn't be acting like this - her feelings, her wants are coming first here, and that's not a good sign!