How to navigate pregnant wife losing job/insurance right before due date

It’s likely that my wife will lose her job about 2 weeks before her due date. Currently, she has health insurance through her company and I through mine. I know I can switch her to mine in the event she loses hers. However, my questions are more around timing and where costs would hit against each of our plans.

For some context – We are both on HDHP/HSA plans. The idea was that it could be cheaper to just pay the out of pocket maximum costs on her plan in the event we had a baby and we didnt have any significant health incidents on my end. Her OOPM was cheaper than a combined plan would have been. It was a calculated risk to take.

She is due the first week Oct. and will need to be induced. So it’s likely that happens at the end of Sept. or the first few days of Oct. but we all know babies are unpredictable little minions and can show up a moment’s notice. If she loses her job in the middle of September, I know her health coverage should cover her through the end of the Sept. I know coverage for her through my plan would start the first day of Oct. But this is where my questions come in.

If she gives birth on 9/30 and is then in the hospital for a few days, how is that billed out to insurance? Is it split by services administered on specific days for her? Is it based on entry date? If by date, I would assume those costs get split to the different insurance plans she is on? Is this going to be a giant headache I’m going to need to histarically vent to the Reddit-verse about?!

I’ve accepted the risk we took with splitting plans will come back to bite me, but I’m hoping to tap the collective wisdom of everyone here on the questions to see what could be in store. If there is a better thread to post this to, let me know.