Nothing that happens to a married couple has any need to be shared with parents. Your mother doesn’t understand her place in their relationship. Your sister needs to distance her family from the intrusive nature of her mom’s meddling. This should be something you can understand as it may well happen to you at some point. For now, your relationship with either should be fine as long as you don’t take sides. Yes, close families do share things like emergencies but your mother has put that closeness at risk by overstepping her boundaries. She is not the primary point of contact for “her baby” anymore.

If it was a minor medical emergency than there is no reason why Op's sister couldn't have picked up the phone and called her mom to let her know about it right away...IF that is what she wanted to do. Clearly that is not what she wanted to do. For whatever reason.

Op's mom flying off the handle like that at her SIL is way out of line.

She needs to be reminded that in the heat of the moment, spouses often have bigger concerns in managing the situation. Also, if the hour is late, if other family is not nearby, or they would be more of a hinderance than a help, notifying them later could be quite acceptable. Your mother needs to step back and know her place.

I also agree that you can support your sister and BIL without doing anything differently with your mom. The argument is between them, not you.

If your mom wants to talk about it with you, you could then tell her what you think in a nice way and then drop it.

My brother and sister were in a huge fight years ago. My sister was absolutely correct in that scenario. My mother kept defending my brother, which in my opinion wasn’t fair. I let her know that while I understood that she loves my brother, it’s ok to admit when he was wrong. And then I said I didn’t want to talk about it more with her.