How to load a new template page according to a particular URL?

You should be able to use template_redirect. I haven’t tested it though, should be something like this.

function page_redirect() {
    if ($_SERVER['REQUEST_URI'] == 'your_url')  {

        require(TEMPLATEPATH . '/mY-account.php');

    }
}
add_action('template_redirect', 'page_redirect');