That is what happened to me.
The graphic on the right was my “friend” for nearly 2.5 hours Tuesday afternoon [from about 2:10 – 4:20 pm EDT].
Thanks to Terri Clevenger, AppleCare iOS & Mac Senior Advisor, and some luck, I recovered and went on to complete the installation.
I learned that others had this problem as well. Apple said:
The issue has been “quickly resolved” and impacted a small number of users during the first hour of availability. In the meantime, if you haven’t done so already, you should back up your iPhone before downloading iOS 10.
Well, I take issue with quickly. Although the journey to solution was made pleasant by Terri, it was a bumpy one. The Update/Restore option was not available in iTunes. After an hour or so, we uninstalled and reinstalled iTunes and selected Restore. That selection failed multiple times. I then lost telephone contact with Terri. Having done so, I selected the Update option [something different–the luck part]. Finally, “the Apple” and a progressing status line.
When it was all done, my apps and data seem all intact. My diligence in backing up was not needed. Terri called 6 times; but, for some reason, my VOIP was not working. I answered her 7th call with my newly recovered iPhone 6S Plus, now with iOS 10, as I was using it to install WatchOS 3. That installation was uneventful and smooth as butter, as it was with the iOS 10 installation on my IPhone 6 Plus. That install took about 50-minutes.
As I said to Terri, I learn little from perfect installations. I am battle ready. Bring on the next iOS10 challenge. Thank you, Terri!