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

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.

Had the same issue. Found that @Billos's 11/2016 answer works...

ctrl-option-shift-down arrow, then left/right to choose user.

However, on my OS X 10.12.2 Sierra system, left arrow/right arrow alone work.

No need for ctrl-option-shift or VoiceOver.