As the person who has just gone through (probably) the most painful event of your life, who is probably still in some pain, bleeding, lactating, trying to breastfeed you can most certainly dictate who comes to visit or not. The "new gran" can wait a week or longer even before she meets the baby, I don't understand this obsession with meeting a baby the second it has crowned? The thing is though if your own parents come to visit, they come to see you and make sure you are ok, if your in-laws come to visit, they are coming to see that new baby, not you.
When I had my third child it was a difficult birth that ended in forceps, I lost a lot of blood and wasn't very well. I ended up staying in for a few days, the baby obviously stayed too but he was completely fine. When I got out of hospital I saw the constant barrage of messages to my husband from my in-laws asking was the baby ok, why is HE being kept in longer, what's going on? It never crossed their minds that it was actually me who was unwell and not once did the messages ask if I was ok! Zero concern for me. Meanwhile my own parent's and sister were concerned about me, they'd been told the baby was fine like my in-laws and realised it must be me who isn't well enough to go home. I didn't actually have any visitors for a week after that birth, I just needed to have some space to recover a little, I felt absolutely dreadful and could barely walk or sit. They are still a newborn at a week old, there is absolutely no rush for people to visit!!
It's your husband's job here to be gatekeeper and manage expectations, I'm so so glad my husband was on the same page as me and completely respected when I felt ready to visit/have people over. I have never had any visitors to hospital except my husband, I'm just not up for that invite every man and his dog over to pass the baby around an hour after you've given birth and have barely had time to shower nevermind attempt breastfeeding without an audience, no thank you.
Never ever share potential baby names before the baby arrives, only share the name when annoucing the arrival of the baby.