In VA I've never failed to take a gun with me when I leave the gun show.
I fill out the paperwork, hand it to the FFL for them to do their part and give them my cell phone number.
The contact the VA State Police (the people/place that do the background checks) and wait for the VASP to notify them it's okay to sell to me. Then they call me and I make my way back through the show to the table, pay and pick up the gun.
We do need two forms of ID here. I use my driver's license and my concealed carry permit.
Many years ago, at a sports store, it took three days to get my Colt Series 80. Some kind of issues with the VASP equipment or methods, or something. Never happened again.
I imagine different states run their states their way. Sometimes an "instant" background check isn't so "instant."