I've just bought an iMac, I have a house full of iPhones/iPads and Windows Laptops and have been using iCloud/iTunes and Apple ID for a very long time, as have my family. So when I started setting up the iMac I wasn't really expecting any issues but I was unable to enter/change or do anything else with Apple ID's - Apple cannot access Apple ID Servers .... it said.
I researched the issue, checked everything I could think of and came up with nada, I tried signing as different people, with different levels of security (lots to nothing) - changed the passwords elsewhere, probably not everything but it felt like it. Then spent 2 hours on the phone with an Apple technician who couldn't understand it either.
Then I had an idea, I disconnected from SuperHub2 and reconnected to my EE 4G Wifi dongle - magic everything works and could sign in. Great got round a very odd problem that is pointing directly at the SuperHub2 (not configured in any particularly clever manner - just tweaked the channels and I use WPA-PSK security until I put my own router back in place anyway, thats another story of broken Superhubs not doing the right thing). The workaround works, sign back into to the SuperHub2 based network and everything is fine, until you need to check your iCloud account and then nothing works unless I disconnect the Virginmedia network and reconnect the 4G EE network.
What is going on; SuperHub firewall, VirginMedia network ?????????
I've had the same problem, and found your thread whilst looking for the answer.
However i've now solved it. The problem is that your IP is appearing on a spam blacklist (google ip blacklist checker) and you'd see it on one (or more) lists. Apples ID server check against these lists and bounce any traffic that comes from them. This is why you can get to other sites ok, and you have to use your 3 or 4G cellular network to login to iCloud etc
I had a similar issue with my Apple ID and I checked to see if my IP Address was blacklisted. I also checked on my 3G connection and the results were similar, with some blacklisting present, so I realised blacklisting of my SH2 was not the problem.
I started to experiment with the SH2 advanced settings. Turning the firewall off didn't help. I then found the solution.
. It was because the MTU (Maximum Transmit Unit) setting on the super hub 3 was too low. I set it back to the default setting of 1500 and all is good.