I am having this issue as well, I had to re-install my Laptop (Windows). I attempted to login but nothing happens, a "please wait" pops up briefly but does nothing else. Creating a new account also produce the same responce within the launcher. I have found what i think is the webfront for the launcher https://launcher.larian.com/login?game=bg3 which also displays the same problem. I attempted to logout of my partners launcher to see if i could login from her MacBook but it appears that the logout function doesn't work either.

I created a support ticket but I haven't had any response. I am hoping they fix this ASAP as I can't access all my cloud saves so I can't progress my game or the game I am playing with my partner. This has been going on for some time, I am a little disappointed that there has been no communication about this as it is blocking access to peoples saved games.