iCloud keychain constantly "Waiting for approval from other device"

Here is what happend - I've restored my TimeMachine backup on a brand new MBP and after that iCloud keychain on it says that it requires approval from another device. What is also happened - iCloud keychains on my other devices (another MBP and iPhone 5s) started to say same thing, so I can not "approve from another device" any one of them. I've tried to activate it using Approval Code multiple times on both MBP and iPhone, but that achieved exactly nothing except "Can not set up Keychain" message.

I've updated OSX on one of MBPs to ElCapitan, hoping, that it will help - still nothing. Rebooting device does not help. Removing iCloud and setting it up again do not help either.

So, no Keychain does not work on any of my devices and I've ran out of options to enable it. Can somebody help, please.