Congratulations!!
I'll speak from my experience on the subject. My wife just gave birth to our first child on Saturday evening, and I am managing an overseas project for my company. I had relocated overseas specifically to manage this project and my wife came with me for the adventure and so that we wouldn't be living apart for a year +. A few months into the project we got pregnant, and it became quickly apparent that the project would not be complete before the birth of our child. This resulted in my relocating back to the US while the project is ongoing (my wife came home pretty soon after we learned we were pregnant, as we did not want to have the baby outside of the US).
My advice on a professional level would be to tell your employer as soon as possible. They should be more than willing to work with you to accommodate (and if not, do you want to continue working for them?). I'd strongly encourage you to go to them with a proposed solution (e.g.: I have a great ASM I can recommend for the show that would be more than capable of stepping up for the time that I'm out) as opposed to just a problem ("I need to miss time"). If you go to them with a solution already in place there's no motivation for them to replace you, and in fact this would only create more work on their part.
Also, it's important that you have a plan that has your ASM always fully in the loop and ready to step up. While due dates are nice, babies are notoriously stubborn about respecting that schedule
. Unless you're planning a c-section or induction on a specific date, you're unlikely to have the baby EXACTLY on its due date.
On a personal level, I'd strongly recommend you take off more than just the day of the birth if you're financially able. Take the time to bond with your new child; they'll only be born once. Work will always be there. And if I can offer one final piece of unsolicited advice (and if you only take one thing away from this post, let this be it): when you're shopping for hospitals, be sure to check out the NICU and make sure it's a facility you'd be comfortable with your child in. You probably won't need it, but if you do need it your baby's life will depend on it - make sure it's a good facility.
