It's usually a lot quicker than that. But since AD has already received an email from PADI, that means his certification has already been processed. The only thing that takes such a long time might be the production and mailing of the card itself. PADI has a number of processing centers, and for example, if the processing center is in one part of the world and the newly certified diver in another part of the world, the time lag increases. Lots of my students live in Europe or North America, and my service center is in Australia, so not only does it take actual card printing time but also depends on the vagaries of the postal systems of two countries. Add to that the possibility of random holidays when the postal systems are overloaded or closed down, and a conservative estimate for receiving cards by mail ends up being 4-6 weeks. Keep in mind that this prevents people from writing to PADI after a week wanting to know where their cards are. If the card hasn't been received by the 6-week mark, there's a good chance it has gone missing somewhere along the line and needs to be reissued. I've had students end up with two cards in this way when their original card failed to turn up on time but was later delivered.
My students and I usually get an email within an hour of when I send in the certification documentation through the online method, even on weekends, and I assume this is what has happened in this case. We already know that a photo for the card was taken on the boat, and we know that AD has not received a temp card yet, so I am guessing that the instructor sent the documentation in online, printed out a temp card, and PADI processed the cert, but that because AD hasn't seen the instructor since all that happened, the temp card hasn't been physically handed to him.
