I think it’s a battlenet app bug. There is a workaround. Find your WoW app and run it manually without battlenet. You’ll find that the Update button turns into a Play button. Dismiss the running WoW app (unless you want to input your credentials manually) and hit Play and you’re up and running.
Unfortunately after you quit the game later, the button will turn back into Update. Blizz needs to fix that bug.