Only nextpagelink on wp_link_pages

You’ll have to prepare your own custom function instead of wp_link_pages to use in single.php file of your template.

Here’s the custom function my_wp_link_pages proposal (contains original wp_link_pages code with commented out lines):

function my_wp_link_pages($args="") {
    $defaults = array(
        'before' => '<p>' . __('Pages:'), 'after' => '</p>',
        'link_before' => '', 'link_after' => '',
        /*'next_or_number' => 'number',*/ 'nextpagelink' => __('Next page'),
        'previouspagelink' => __('Previous page'), 'pagelink' => '%',
        'echo' => 1
    );

    $r = wp_parse_args( $args, $defaults );
    $r = apply_filters( 'wp_link_pages_args', $r );
    extract( $r, EXTR_SKIP );

    global $page, $numpages, $multipage, $more, $pagenow;

    $output="";
    if ( $multipage ) {
        if ( 'number' == $next_or_number ) {
            $output .= $before;
            for ( $i = 1; $i < ($numpages+1); $i = $i + 1 ) {
                $j = str_replace('%',$i,$pagelink);
                $output .= ' ';
                if ( ($i != $page) || ((!$more) && ($page==1)) ) {
                    $output .= _wp_link_page($i);
                }
                $output .= $link_before . $j . $link_after;
                if ( ($i != $page) || ((!$more) && ($page==1)) )
                    $output .= '</a>';
            }
            $output .= $after;
        } else {
            if ( $more ) {
                $output .= $before;
                /*$i = $page - 1;
                if ( $i && $more ) {
                    $output .= _wp_link_page($i);
                    $output .= $link_before. $previouspagelink . $link_after . '</a>';
                }*/
                $i = $page + 1;
                if ( $i <= $numpages && $more ) {
                    $output .= _wp_link_page($i);
                    $output .= $link_before. $nextpagelink . $link_after . '</a>';
                }
                $output .= $after;
            }
        }
    }

    if ( $echo )
        echo $output;

    return $output;
}