I think just about every airline will try and get you back home ASAP if you're on the road and your wife goes into labor.
My wife and I had our first child last year. I bid my vacation for the month she was due. I also bid to front load my schedule the month before and back load my schedule the month she was born since the due date was towards the first of the month. I was fortunate that everything worked out and she was born a few days early during my off time and I was able to spend nearly a month off.
Another option is to look into FMLA for time off. I'm not sure if it's the same for everyone, but where I with only get paid if the reason for FMLA is for yourself and not for taking care of someone else. It's easier said than done, but don't worry about money if you take this route and aren't paid. Make sure to spend time off as a new family and the finances will work out in the end. Congrats.