Yeah, I’m having the same problem today as yesterday.
@Grahran Are you seeing WOW51900123 as the error code?
So the problem still persists, I had a lot of trouble due to removing and then readding authenticator, too many codes sent on phone, had to ask a GM to remove my phone number even, now I was able to add the authenticator back but indeed still doesn’t get authentication requests by wow itself, can only log with launcher and close it after.
I get the same error code as Apricot, above! 
Thanks also to Game Master Ingrothar who kindly pointed me also to this thread. I am confident they will find a solution for this, soon.
I heard there was a 15 mb patch in the us that fixed this problem for some people, but in the eu it still asks me for the authenticator code, despite downloading a 100 mb patch or so, is the problem still occuring in eu for you?
Still happening just now.
Somehow I’ve managed to work around it. It’s an annoying work around but at least I can login.
At the login screen if you have an email address change it to your phone number. You will still get prompted for the authentication code but it works. The next time you login, if the phone number is in the email box you have to switch it to your email. So just alternate between them.
It looks like this prompts the account selection window which is, again, another extra step. But at least you can get in.
Hope this helps and I hope it gets fixed soon.
Edit: As a note, I have 2 WoW accounts on this BNet account. Not sure if the account selection screen pops up for single accounts.
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Now works for me in EU Retail… But TBC remains unfixed.
Fixed for retail, thx Blizz 
-edit-
The popup still comes, but if you manually enter the code, it seems to work now.
I too can confirm this has been sorted on Retail WoW. Well done Blizzard for getting this resolved quickly! 
Ah, that’s fine, before basically no code was being sent, so using the manual didn’t match ofc.
I can see some guys are confirming this issue has bean fixed for them. This stopped working for me at exact same time as reported above but still doesn’t working for retail. I even tried removing the authenticator app and adding it again but without help. Maybe the fix is not applied for all of the European realms yet.
Quickly? Eh, it’s been several days.
But yes, it seems to be working now.
@Stravà: There was a small patch required before I could attempt to log on.
Indeed, you just reminded me we have to launch battle.net app from time to time to trigger client update since game is not smart enough (anymore) to tell you there is a new client version. Thank you @Cataracts, it’s working now 
Still not working since the patch on the 19th, so removed my authenticator in order to manually log in again, what a joke.
Ye, where’s the fix everyone’s talking about? The update for battle.net aswell?
Still not working for me.
Hi All,
I can log onto WoW Retail using the .exe file, as other posters above it still asks for the authenticator code but with no further issues. However, I still can’t log onto Classic or Burning Crusade.
My ticket is still active so I’ve just let them know that its still an issue. Lets hope that any maintenance on Wednesday will resolve it but maybe I’m being a bit optimistic 
Still not fixed for Classic TBC.
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As mentioned, issue is ongoing for Classic TBC.
But the workaround method with alternating email and your phone number works.
But it requires your authenticator every time which wasn’t the case previously.