You've got two instances of the jQuery script on the page! This can create a lot of conflicts and confusion. Either you or one of your plugins is linking to jQuery without using the proper wp_enqueue_script
function.
I'm gonna go ahead and point the finger at Advanced Ajax Page Loader. Try deactivating that plugin and see if your sliders start working. I've looked into it, and that plugin actually does a decent job of enqueueing the script:
function enqueue_AAPL() {
//Make sure we use latest jquery?
wp_deregister_script('jquery');
wp_register_script('jquery', plugins_url( 'jquery.js' , __FILE__ ));
wp_enqueue_script('jquery');
}
So my guess is that you've included <script type="text/javascript" src="http://www.astwood.co.uk/testsite/wordpress/wp-content/themes/AstwoodV3/js/jquery-1.8.2.min.js"></script>
somewhere in your header file. Find it, take it out. Then make sure you're enqueueing all your scripts (like jQueryUI, fancybox, liquid-slider, jquery.tweet, etc.) keeping in mind the dependencies
parameter