This thread explains my update experience. When I clicked to install iOS 7, it took barely ten minutes to install. I happened to be watching my ADSL modem as it did this, and it never once blinked, which was puzzling because that meant that nothing was being download. Nevertheless, after ten minutes passed, iOS 7 was up and running.
I now realize that the phone must have downloaded the entire update in advance, the first time it popped up a message asking if I wanted to update, and it just kept the files on the phone even though I said no. When I finally chose to update, it used the files it already had to do the update, making it extremely fast.
I wonder what happens on phones that don't have space to download the update (although I guess then they don't have space to do the update, either). And if you run out of space after the update is downloaded, does the phone silently delete the files, wherever they are? They aren't counted in usage statistics, as I would have noticed a 700 MB download in the numbers.