When I had my Green Card, I was always delayed. Apparently they had to call immigration and manually clear it with them. I also had to show my Green Card along with other documents. However, it never took more than a couple of hours. I think the fastest I've ever been approved was when I bought my AK-47 - it only took about 30 minutes and the longest took about 2 hours. It's funny that AK was so fast out of all guns, since it's the most "evil" looking one. I guess since I'm Russian it went through quickly. ;-)
Nearly all my purchases were done at a local store, on Saturdays, and I've never been delayed to the next day. I already had a routine down when I go in, inspect the firearm, fill out the paperwork, go out for lunch and come back to pick it up. I knind of expected it to be always the process for me since I wasn't born here, so imagine my surprise when I came in to by my first gun since I became a Naturialized Citizen and was just ready to head to lunch when they approved me on the spot. Ever since then I've been approved almost immideately nearly every time. I got delayed once, but that was on Saturday, at the very end of the gun show, and the delay was only 15-20 minutes or so.
Being an immigrant, I have a fairly rare name for the US and I also put my SSN down, so maybe that speeds up the process as well. But, I guess you never know. I don't understand why I get approved faster than some natural born members of LE or Military with a security clearance. I'm not complaining, but it doesn't seem fair to me...