I have had 2 children.

I'm sorry that happened to you. I had my mother and mother-in-law in the room for my second birth (they didn't make it in time for the first) and it was really nice. They held my son and admired him while I got cleaned up, had IVs removed, etc. They both told me later that it was one of the best experiences of their lives and they were grateful to be there. They both have several grandchildren but my son was the only one they got to see born. They're very sweet, respectful women, though, and I'm sure that makes a difference.

No one should be pushing their way in if OP isn't ready. But I'm not sure why baby can't be taken to the nursery for a moment (if the birthing center has a new baby viewing area) or OP's husband can't take some pictures and then walk out to show his relatives. I come from a close family, too, and we always filled up waiting rooms when babies were born. It wasn't about trying to get in the room (usually we didn't even go back past the nursery viewing window) but it's just about being there. I guess it's hard to explain if it's not what you're used to.

No matter if OP is justified or not, her husband's family is going to feel very left out and on the outside of things if they can't even be in the waiting room or see the baby for several days while OP's mom gets to be there for the birth.