I can't speak for LLC situations. But, if you buy from a dealer, once he either has your cash or you've got an approved loan, the dealer prints the DMV (aka DVLA) documents and you drive away. Hopefully, you'll have done the walk-through and got things fixed before driving away. Getting the stuff fixed might take a few days. Plates will follow in the mail to whatever mailing address you've provided.
Don't forget to line up the insurance ahead of time, and allow time for PDI and orientation.
FWIW we'd been here less than 2 weeks, living in a motel, and on UK payroll (i.e. we were on a B1/B2 visa) when we bought our first car here. We didn't have insurance pre-arranged, so I had the dealer drive the car on their plates. I picked up the phone book and an insurance agent met me in a restaurant at 10.pm to write the insurance binder. Just like your friendly Pru agent ;D
The longest part of the vehicle buying process is arguing about the price, but that can be shortened by walking out if they don't accept your offer. Start your stopwatch and see how long it takes them to come after you.