I had this problem this morning. I think it was because I hadn't updated my IOS for sometime. Spent 2 hours on chat to Applecare who told me to restore from itunes and then restore from icloud on phone, which I tried 3 times but it did not work.
I hadn't backed up my phone since July unfortunately. So I would have lost everything anyway. But each time I restored it allowed me to go through completely as if it were finished and then again would stick on the 'slide to upgrade' screen.
Anyway I have now worked it out for myself!!
1. Plug into computer with upgraded itunes.
2. Whilst on the 'slide to upgrade screen' hold power and home button until the itunes symbol appears with the lightning cable below on the iphone screen.
3. Click restore on itunes
4. Then select update and restore.
5. After this, go through as normal and then select SET UP AS NEW IPHONE
6. DO NOT ATTEMPT TO RESTORE FROM ICLOUD BACK UP IF YOU HAVE ALREADY TRIED THIS it will not work!!
7. If you set up as new iphone you can log in on old apple account etc. The phone is as new, only my contacts saved (thank god) but it works
8. I left it at that because I would have lost all recent photos anyway due to not having backed up for along time.
9. However, you could now attempt to restore from icloud through settings in the usual way as your iphone should now be working.
Hope this works for everyone as it did for me. All I know that by restoring it through itunes and then attempting to restore from icloud during set up by phone went through the whole set up and then went straight back to the slide to upgrade screen. Setting up as a new iphone is the only thing that worked.
Hopefully apple will fix it soon, I think it is appalling and it has wasted my whole morning. But having to re-install apps is a lot better to having an iphone stuck on one screen.
Also, attempting to wait on the slide to upgrade screen for it to start working will also not work!! I waited for hours and tried everything, this is the only thing that would work.
Appalled that this is the 4th iphone I have had in 2 years and nearly lost this one two. Still wouldnt use anything else tho!
Let me know if this works for anyone.
Jem x