How can I choose a user an the login page without a mouse?

As a old-school Unix junkie I don't like being forced to use a mouse when a simple keyboard shortcut would do. Today I found myself needing to login to an OSX machine just to launch an app, but the system's mouse had a dead battery. I tried about every key I could think of on the keyboard and could not figure out how to select a user from the login screen. I could change the system volume and any number of other useless actions, but a keystroke that would change the input focus to the user selector was beyond my ability to guess.

Even more frustratingly, when I did finally hook up a mouse, merely moving it seemed to bring the login window into focus and the arrow buttons could be used to select a user.

On attempting to duplicate this later, it seems the initial login screen does have keyboard input focus and works on first bootup. However if you either wait a minute or two for the mouse connection dialog a dialog pops up wanting you to connect a mouse. This steals the input focus and never gives it back to the login screen. If you do not connect a mouse or if an almost-dead wireless one connects and disconnects, the login screen cannot be accessed from the keyboard.

Is there a key-binding I am missing that will re-focus the login page? Or is one just stranded at that point? Is there a keystroke that will reboot the system so you can catch it before the mouse connect dialoge pops up?

The clue was from activating the voice-over, which eventually tells you:

Ctrl + Option + Shift + Down Arrow simultaneously followed by left/right to choose user,

You don't need to activate voice over for this to work.

That is completely non-intuitive: a simple tab would solve this completely.

It's certainly not an elegant solution but you can use the VoiceOver screen reader to accomplish something similar - at least to move the focus. VoiceOver is used by blind people to work the computer without a mouse at all and by using voice output and keyboard shortcuts.

At the login screen, press Command+F5 to enable VoiceOver. Your computer should start talking - and also display what it is saying in a small box in the lower-left of the screen. Try pressing Control-Option-Left and Right arrows to move the focus around. You should be able to select the different user accounts. Once you've selected the one you want hit Enter, then you can switch VoiceOver off (again Command+F5) and type your password.

If a window is in the way, you can use the Window Chooser to move focus. Normally this pops up an on-screen view, but apparently at the login screen it is voice-only. Press Control+Option+F2 twice rapidly, and you should hear Window Chooser. Use up and down arrow until you find Login window, then press Enter. Now the previous commands (Control+Option+Left/Right) should work.

VoiceOver is a complex system so explaining it in any detail is out of scope for this question, but there are also ways to get rid of that mouse connect window.