I had the same issue everyone else is having. Couldn’t log into the Battle.net desktop app as it was placing me in an infinite queue.
My work around was logging into the battle.net mobile app and enabling authenticator in the menu. If you were successful, you should close out of the Desktop Battle.net app and try logging in again. This should prompt you for a code or for you to hit “approve” on your battle.net mobile app
If you are having issues logging into the battle.net mobile app, try clearing the storage/app cache or try reinstalling it. I hope this helps. Good luck.
Edit: There is one other thing I did that may have helped me log into the mobile battle.net app. At first I didnt think it helped or maybe i was just lucky with the log in. I downloaded the old mobile battle.net authenticator and tried running it. It directed me back to the new mobile battle.net app. However, I went back to the old authenticator and continued setting it up. It got stuck on “setting up” but I went to the new battle.net app and I was logged in. Still not sure if this helped me log into the battle.net mobile app, but it was one of the steps I used. Maybe I just got lucky. Best of luck to you all and let me know if it worked for you.