You probably won't be contacted, and I don't think checking will speed anything up. If you get a bad background check, you'll probably get a copy of the letter to the court. If you don't, you may not hear anything until the permit actually shows up in the mail.
That's what happened with my wife's permit last year. We were on our way to Rehoboth, and picked up the mail through the window of the car. I handed her my backup gun, and off we went.