I have recent personal experience in this regard, but I don't know how generally applicable it is.
When I contacted the insurance company to ask about this in advance, they told me everything that happened at the hospital under the initial admittance would be billed to the first insurance, but that if anything happened after discharge that required a new admittance, that would be on the second insurance.
That turned out not to be true. For most of the line items, that was true, but not all charges ended up under the same general admittance, so they had individual dates that were different from the admission date. There was one charge that, despite the date being on the delivery date, was billed to the second insurance. When questioned, the insurance company confirmed it should be on the second insurance. I might have been able to fight this, but didn't because I expected to hit the deductible on the second insurance even without that charge.
Additionally, some charges were separate for baby. Since baby was never on the first insurance, these were then billed to the second insurance.
The total of the charges that were billed to the second insurance was about equal to the deductible of the second insurance (while the total billed to the first insurance was significantly more).
Something else to consider: when I say things were billed to the first vs second insurance above, that is a convenient lie for the sake of simplicity in explaining which insurance ended up covering which things.
In reality, everything was billed to the first insurance policy first, which then reviewed all the charges and rejected the ones it decided were not its responsibility. I then needed to contact the hospital to have those charges re-billed to the second insurance.
If the hospital and insurance company are both reasonable and cooperate, this isn't a particularly herculean task, it just takes time and mental energy. But insurance companies are insurance companies, with policies that can be difficult to understand, and may not be consistently applied, so you might end up in a situation where both insurance policies reject a charge, and then it's up to you to argue with them about it, which can take significant time and energy, which you probably don't want to deal with when you are contending with a newborn. Depending on how long it takes to get such an issue resolved, there may be financial implications for you too, with you either needing to pay the bill while the insurance is sorted, or be considered delinquent on the payment.
Personally, I was prepared to pay the full deductible on both insurance policies. Due to the various costs and incentives involved with the two plans, it was a calculated risk with large upside and small downside, but you will need to do the math for your own situation.