The application process can be done in 1 day, though is more usually over 2 days, as the appointment for the interview at the registration office isn't usually made until all the paperwork is collected. It's at the interview stage that you know whether you've been accepted or not.
The card is sent to your Turkish address & friends recently got their cards within 2 weeks of application. Note, you get a text that it's on it's way to you &, if you're not home, when it's delivered you have 7 days to collect it from the PTT by the otogar.
You can leave Turkey before the card is received if a) your tourist visa is still valid or b) you get your application form stamped by the kaymakan's office, in which case you may only leave Turkey for 14 days.