This error is related to the AADC restrictions with accounts registered to the United Kingdom. You may need to check your Privacy settings via the link below and make sure everything looks correct there.
I had this exact same issue.
The solution which worked for me is on the account privacy settings - I had to turn on RealID.
After that I could log in and get in the game.
When I turned it off, logging into the Battle.net launcher and Diablo 3 EU server still worked.
So in short, if you had RealID turned off before the AADC regulations took effect, you have to turn it on. You can then turn it off again.
Shame none of the 3 GMs that replied to my support ticket knew or suggested that…