Resolved:
Apparently the client failed to patch, which was why the game couldn't login. Closing and reopening the client doesn't show it as downloading, but rather as 'Complete'.
So what I did was click the cogwheel icon, click "Check for Updates", where I soon got a popup that said 'Error 103 : Already Downloading'. Apparently it was downloading already without the UI displaying that it was downloading.
I closed the client down again, reopened, and there it was... blue bar moving, it was downloading. Patched, and logged in. Problem solved.