I required this code in the Wordpress codex which shows Child pages from the current page in publish format.

I produced a webpage known as "Home" and designated a template known as Top Of The Page into it:

front-page.php:

<?php
/**
 * Template Name: Front Page
 * @package WordPress
 * @subpackage Twenty_Ten
 * @since Twenty Ten 1.0
 */
 get_header(); ?>
 <div id="intro">
  <div id="tagline">
   <?php // Retrieve a list of latest posts or post(s) matching criteria).
   $args = array('category_name' => 'Tagline', 'numberposts' => 1, 'order' => 'DESC');
   $customposts = get_posts($args);
   foreach($customposts as $post) : setup_postdata($post); ?>
    <h2 class="entry-title"><?php the_title(); ?></h2>
    <div class="entry-content">
     <?php the_content(); ?>
    </div>
   <?php endforeach; ?>
  </div><!-- #tagline -->
  <div id="featured">
   <img src="<?php bloginfo('template_directory'); ?>/images/<?php echo get_option(THEME_PREFIX . 'intro_image'); ?>" />
  </div>
 </div><!-- #intro -->
 <div id="main-content">
  <div id="main-content-first">
   <?php // Retrieve a list of latest posts or post(s) matching criteria).
   $args = array('category_name' => 'Main Content First');
   $customposts = get_posts($args);
   foreach($customposts as $post) : setup_postdata($post); ?>
    <h2 class="entry-title"><?php the_title(); ?></h2>
    <div class="entry-content">
     <?php the_content(); ?>
    </div>
   <?php endforeach; ?>
  </div>
  <div id="main-content-middle">
   <?php // Retrieve a list of latest posts or post(s) matching criteria).
   $args = array('category_name' => 'Main Content Middle');
   $customposts = get_posts($args);
   foreach($customposts as $post) : setup_postdata($post); ?>
    <h2 class="entry-title"><?php the_title(); ?></h2>
    <div class="entry-content">
     <?php the_content(); ?>
    </div>
   <?php endforeach; ?>
  </div>
  <div id="main-content-last">
   <?php // Retrieve a list of latest posts or post(s) matching criteria).
   $args = array('category_name' => 'Main Content Last');
   $customposts = get_posts($args);
   foreach($customposts as $post) : setup_postdata($post); ?>
    <h2 class="entry-title"><?php the_title(); ?></h2>
    <div class="entry-content">
     <?php the_content(); ?>
    </div>
   <?php endforeach; ?>
  </div>
 </div><!-- #main-content -->

<?php
 $mypages = get_pages('child_of='.$post->ID.'&sort_column=post_date&sort_order=desc');
 $count = 0;
 foreach($mypages as $page)
 {
  $content = $page->post_content;
  if(!$content)
   continue;
  if($count >= 2)
   break;
  $count++;
  $content = apply_filters('the_content', $content);
 ?>
  <h2><a href="<?php echo get_page_link($page->ID) ?>"><?php echo $page->post_title ?></a></h2>
  <div class="entry"><?php echo $content ?></div>
 <?php
 }
?>


 <?php get_footer(); ?>

I Quickly made the About page its child (Home page's child).

For whatever reason there is nothing being displayed (see at the end from the template)

Any suggestions?

I am speculating your problem is the fact that $publish is designated within the loop, which code is positioned outdoors from the loop.

The issue could just be the main one demonstration of $publish->ID your page's structure is fairly strange. Use this line rather than the same one you published:

$mypages = get_pages('child_of='.$wp_query->post->ID.'&sort_column=post_date&sort_order=desc');