I had the same issue as everyone else, and called Apple Support saying that I have been unable to access iMessage and FaceTime for over a week and I read on some forums that it might be that I've been blocked out for some reason. The guy I spoke to took some info from me and then directed me to a senior technician who (after taking some security questions) unblocked it for me in about 15 minutes! I restarted my computer and it works fine now!
I don't know if I just got lucky or if it was because I knew about being blocked out - but asked very nicely about it - but this is the fastest I've had an issue like this fixed! He did say it was a once off to be unblocked, and couldn't tell me WHY I was unblocked, though... that would have been nice to know to avoid doing it again..
All I can say is contact Apple Support and tell them exactly what happened (and it pays to be nice about it), and hopefully they'll unblock you.
Good luck anyone still suffering!