Query by key or author

You don’t really need to use the &new WP_Query just new WP_Query.

The following code will resolve your problem.

add_filter( 'posts_search', 'q166788_search_modify', 10, 2 );

function q166788_search_modify( $sql, $query ){
    // Only if our variable is true we do the black magic
    if ( ! isset( $query->query_vars['search_conditional_toggle'] ) || $query->query_vars['search_conditional_toggle'] !== true ){
        return $sql;
    }

    return preg_replace('/AND/', 'OR', $sql, 1 );
}

// The actual search
    $s="bord";

    $args = array (
        'author' => '1,2,3,4,5',
        's' => $s,
        'search_conditional_toggle' => true, // Make the black magic happen
        'posts_per_page' => -1
    );

    $allsearch = new WP_Query( $args );

    var_dump( $allsearch );