We applied for passports for kids on Nov. 5. They are already on the mail and scheduled for delivery the end of the week. So less than a 3 week turnaround.
Also, unlike with our previous application, the processor at the USPS certified a photocopy of the birth certificates and gave us back the original. So we were able to keep the birth certificates.
Our state has an office for walk in requests for same day birth certificates. By mail only takes a couple of weeks here.
Also, the passport application asks when you'll be traveling, and they will supposedly try to get it to you on time if you submit the application in accordance with the processing time guidelines for travel posted on the department of state website.
Bottom line, look into getting a birth certificate replacement, so you know what you need to do just in case the department of state let's you down, but get that application in for the kids passports this week if possible.
Fyi either both parents need to be there,OR the absent parent can send a notarized form giving permission for the kids to get a passport. Department of state site has the form. Appointments are a lot easier to get during the week (here, next day appointments are always available at the post office, but Saturday appointments can be over a month out), and as it is often hard for both parents to get out of work at the same time, this form can make things a lot easier, even though getting it notarized is an extra step.