Probable Cause

I strongly suspect that you do not call wp_head() in your header.php file, immediately before the closing HTML </head> tag:

<?php wp_head(); ?>
</head>

You must call this template tag, in order to fire the wp_head action. The wp_head action in turn fires the wp_enqueue_scripts action. So, without having <?php wp_head(); ?> in your template, you won’t fire wp_enqueue_scripts.

Syntax Errors

You have at least three syntax errors:

1. Concatenated script URL:

   wp_register_script('easing', get_template_directory_uri() . 'http://gsgd.co.uk/sandbox/jquery/easing/jquery.easing.1.3.js', array('jquery'), '1.3', true);
   

   This:

   get_template_directory_uri() . 'http://gsgd.co.uk/sandbox/jquery/easing/jquery.easing.1.3.js'
   

   …should be this:

   get_template_directory_uri() . '/js/easing/jquery.easing.1.3.js'
   

2. Array key string:

   wp_register_script('jquery-mob-custom', get_template_directory_uri() . '/js/jquery.mobile-1.0rc2.customized.min.js', array(jquery), true);
   

   This:

   array(jquery)
   

   …should be this:

   array( 'jquery' )
   

3. Incorrect wp_enqueue_script() parameter ordering

   wp_register_script('hoverIntent', get_template_directory_uri() . '/js/jquery.hoverIntent.minified.js', array('jquery'), true);
   wp_register_script('jquery-mob-custom', get_template_directory_uri() . '/js/jquery.mobile-1.0rc2.customized.min.js', array(jquery), true);
   wp_register_script('home-page-slideshow', get_template_directory_uri() . '/js/diapo.js', array('jquery'), true);
   

   In each case, you need to add the $version parameter between $deps and $footer