You can get on the PTR and learn to love patch 2.4 by heading over to the Account Management page on the main World of Warcraft site. When there, click on the "Test Realm Character Copy" button, and then on "Copy a Character". After you've asked Blizzard to copy your character over, come back and check about every hour to see if they've done it yet. Even though the wait time says four days, in my experience it has been substantially less.
One last tip: if the button comes up gray, just hit refresh a few times.
Updated 1:15 p.m. EST: Screams of frustration can be heard around Azeroth as Blizzard plays dirty tricks with their character copy process. People are now having mixed results in getting their characters copied across. It might be that the queue is full again.