I don't have kids, so I'm missing any experience on that whole side of things. I don't have any opinion on how you "should" feel about it, and my perspective on it is that of someone who's never been in anything like your situation, so it can only really be about a possible viewpoint of the some of the other people in this whole thing.
I don't want to offend with my own perspective, as someone who's never experienced this situation but does have a family… this isn't meant as a comment on anything you might be feeling or would have wanted. It's more about imagining how I might feel as a family member, in the context of my own family situation.
If I had a sister, or even a cousin I was close to, and she had had a difficult labour resulting in (what sounds like) emergency major surgery, and possibly there had also been worries about my niece or nephew's life/health being at risk at some point, but it all turned out okay… well, to be honest being told only after a long delay of many hours or even some days might feel weird. Only finding out much, much later that a serious emergency had been happening for a member (or two) of my family, people I love had potentially been in danger, it had thankfully resolved well and happily, and I'd been oblivious to it all — when a family member knew about it, was in the WhatsApp group, and could have told us all straight away that this had happened but all involved were now doing well — it's… an unfamiliar situation. It might feel strange to look back on a conversation I'd had with the person who knew, in those hours or days afterwards before you shared your news, when they knew you'd been through hell and come out okay, but hadn't told anyone.
I absolutely know that I wouldn't be entitled to that information about somebody's health, but any other situation where a family member had a worrying medical emergency, had surgery and was now doing well etc., it would seem a bit odd to keep the whole thing from family for potentially hours or days, until the person involved chose to share.
But I do realise that giving birth isn't "any other situation", though, that there are big differences emotionally, culturally, relationally, socially, etc., and that people have really strong preferences about things like delaying announcements until they feel ready, choosing when/where/how/to whom they'll share, and so on.
I think maybe in a situation where there had been risk of things going wrong, in your MIL's position I might just feel an urge to share with family as soon as it's reasonable to do so — if not when the emergency is happening, then at least as soon as I know that everything's okay. A rush of relief, perhaps — releasing all that worry by sharing the fantastic news that everyone's okay. The trouble is, that doesn't mesh well with your autonomy, and your right to be the one sharing your big news when you're ready. It's a tough one.
I'm sorry it all feels like it's been taken out of your hands — I hope that the news-sharing was entirely without malice, and your relationship with your MIL isn't badly damaged.