Looks like we are a little ahead here in the north
You go to the city office, pull a number, wait about 10 minutes (yes, I'm serious, they're actually efficient here!), get a desk number, go to the clerk, hand him your papers and the old number plates (if you have a secondhand car), he reaches under his desk, pulls two ready-to-use new number plates out, puts the T?V stickers on, you say bye, pay the fee at an automate using your cash card and off you go. Took me 20 minutes last time.
Obviously the city office has some number plates printed in advance by those private businesses. You only actually have to go there, if you need a special size or if you want a customized plate (which costs extra, too).