So after more than a month of not being able to connect to the servers, I may have a fix. So after going through my whole system for the 55th time, I decided to start the Blizzard app-run as administrator which was a random suggestion as a fix. Well blow me down sailor, I was able to actually connect to the servers and play my campaign.
I tested it by exiting the game and the Blizzard desktop app and doing the normal click the app and try to play the game and it wouldn’t connect me to the servers.
Exit game and app, run as administrator again and I got in no problems again.
Blizzard clearly has an issue with the desktop app that they need to fix.
give it a try and see if it helps. Win 10-64bit.