Can a plugin be run in a different language than WordPress?

Assuming the plugin makes use of localization at all, you could for example try exchanging the possible load_plugin_textdomain call with load_textdomain where you can point to the file you like.

A cleaner way would be using the plugin_locale filter that let’s you modify the locale used by a plugin:

function my_plugin_locale_filter($locale, $domain)
{
    if($domain === 'the_plugins_textdomain')
    {
        return 'cy_CY';
    }

    return $locale;
}
add_filter('plugin_locale', 'my_plugin_locale_filter', 10, 2);

That way the plugin can more or less safely be updated without the custom code being overwritten.