I've spent some hours trying to override updown module templates in a theme till discover that it just doesn't work as expected.
I copied the .tpl.php files to my theme directory, customized them and cleaned theme registry but drupal didn't recognize those files and continued rendering the content using the original module template files.

After some code investigation I discovered that the hook_theme registers the updown_active_widget and updown_inactive_widget theme functions pointing respectively to updown-active.tpl.php and updown-inactive.tpl.php. But when drupal searches for theme templates it expects to find updown-active-widget.tpl.php and updown-inactive-widget.tpl.php files. So if one copy the module .tpl.php files to a theme directory, drupal will not find these files and no override template module is possible. To workaround this, just rename the .tpl.php files copied to theme directory as drupal expects. And to fix this definitely, the hook_theme function should be altered in order to respect drupal name patterns.