When an update is first announced, millions of users rush to install it - and Apple's servers get bogged down. Even with an Ethernet connection to my router, I've had an iOS download take more than 24 hours! Some updates are huge (~4GB), and those can take quite a while, especially over a wireless connection.
It's better to update via a Mac, if you have one. Once the download is in progress, you can disconnect the phone and use it as usual. Eventually, maybe the next day, you'll plug it into the Mac and discover that the update is ready to be installed.
If you have a friend who's done this, I think you could attach your phone to his Mac, go through all the security rigamarole, and use the already-downloaded update.