If your husband has more than two dependents (you and the 2 YO), he must get a waiver for any additional dependent (the new baby). This could be a delay of six to eight weeks, and he could lose any contract he has signed and need to start over in the DEP process. Fraudulent enlistment if they want to push it (which I have never heard of happening, but potentially on the table). Technically though, the baby does not count until you have deliver.
Any questions about having a baby during different phases of active duty? Boot camp, A school, in the fleet, during deployments? BTW, paternity leave is not for the birth, but for a period during the first year after the birth. I can give details if you;d like to know how that works.
He probably would not lose his contract, but in my personal opinion, I'd say wait until after A school would be ideal. The reason is pregnancies can be tricky in those first few months, and you'd not have much access to him were anything to happen. Also, starting with one OB-gyn and then switching docs can be frustrating, and moving while pregnant is just not pleasant. Another reason, just that much less to worry him through boot camp.
If you're already pregnant, just don't say a thing to anyone. That includes here. Gossip always finds a way to get to the wrong ears! (or eyes, in the case of letters)