How to fix iCloud "cannot connect to server" / changed Apple ID
I had major problems with iCloud "cannot connect to server", and not being able to sign into iCloud even though I'd used iForgot to reset my password. It seems the Apple servers got confused coz I'd changed my Apple ID (email address) at some point.
So anyway what I did was, started a Live Chat Session with Apple support, and the girl took me through some really strange/unintuitive steps, and reset a few things at their end, and it works now.
Basically, it's not possible to fix yourself! Live Chat is the way forward.
Hope this helps someone