There will be a delay of a few days, but if you play your cards right you could get some of PLT benefits early, even minutes after arrival of the qualifying flight. The way to do this is by calling AA Customer Service (NOT the regular AA reservations line) and ask them to notate your PNR to the fact you have just qualified. If they agree you are set, if they don't you just have to wait for the system(s) to recognize your status.