I know Pat Fulwiler (she is a an attorney.) I'll be very surprised if she doesn't give you a complete response.
I was under the impression that the initial plastic card came from Albany and the county had no control over the time frame. Perhaps I'm wrong. If the cards are printed locally then there is no logical reason why they can't be done on the spot.
I recently had a change of address and a new card was printed and given to me in less than 10 minutes.
BTW - when I got my permit it took 12 1/2 weeks to receive the card in the mail.