When it finishes downloading it will typically restart itself and run the update within a few minutes. I don't really know how to tell if it is downloading other than the spinning icon in the upper-left screen, but if your connection is good it will download the update in a minute or two, so you might not notice the download taking place. I think the spinning icon only displays at the end of the download, so if you blink you will miss it.

If you think it might have downloaded, you can go into Settings and select Update Your Kindle from the menu. If Update Your Kindle is greyed out then the update wasn't downloaded.