Another question about flying with a baby

Not sure if anybody's had any experience with the new registration requirement (ESTA) when flying to the US. We've already booked our tickets and when it came to stating the nationality and passport number of the passengers, I put British for my baby as I haven't gotten his US passport yet. I know my husband has to do the ESTA thing, but will I need to do one for my baby? He'll have his US passport by the time we go, but his ticket information will show him as British, not as a dual citizen.

And a related question - how long will it take to get his US passport once we've applied? I've just realised I might not have left enough time if it isn't a quick process. We fly to the US on 23 April.

Many thanks

I just found the answer to my question about how long it takes to get the passport. The consulate website is stating about three weeks so we should be okay.

About the passport, the consulate told us to expect in within 3 weeks also but it came exactly a week after our appointment.  I know others on here have posted receiving there's in a similar time.  This was in Sept. for us.