Internet not working on iPad mini with iOS 7
I solved it - turns out the problem was a flaky internet connection! The clue was that I was unable to sign in to iTunes on my Windows laptop (the connection was timing out), even though I could do the same on my Linux laptop.

I rebooted the router and then was able to sign in to the iCloud account from the iPad and everything started working. I also updated it to iOS 8.4 just in case :-)