A local AASP took 4 days to get the warranty parts and install them. It just really depends.
[Not rant]Honestly, I just back a MacBook 2008 that went in for a top case replacement and screw replacements (I paid for all the missing screws). The AASP forgot to put in 2 screws, which would have been find if they were free. Plus, they misaligned some ribbon cables that are now blocking the RAM slot and put in some of the screws in the wrong places. I hope the Apple Store can fix it.[/Not rant]
That's because the Apple Stores normally have quite a few on hand. Both MacBooks I've taken in to the Apple Store were done in less then an hour. I was barely even able to leave the mall parking lot!
Well its good to know Apple stands behind their parts and AASPs.
[Not rant]Honestly, I just back a MacBook 2008 that went in for a top case replacement and screw replacements (I paid for all the missing screws). The AASP forgot to put in 2 screws, which would have been find if they were free. Plus, they misaligned some ribbon cables that are now blocking the RAM slot and put in some of the screws in the wrong places. I hope the Apple Store can fix it.[/Not rant]
Seriously though, find a reputable AASP and you should be fine.