So that is a special circumstance that you may be able to work around. Depending on the timing, you may be allowed an emergency leave to be there for his birth. The Navy does what it can to accommodate families. It can't always, but it tries.
Have you brought up the timing with your recruiter? Maybe you can go through them to get in touch with your A-school and try to arrange things. Of course this all assumes you do not get held back in boot camp.
There is a pretty good chance you will be in holding at A-school for a week or so when you first get there, meaning that you would not miss out on much. You could also offer to voluntarily take supplementary class time until you catch up, which some A-schools also do.
The point is, the worst they can do is say no. Which it sounds like you are already expecting to hear, so it wont be a big let down
edit: I do not mean "emergency leave," which is a specific thing. I mean "emergency" leave as in short-fuse quick turn around.