Replace custom gallery shortcode with default gallery shortcode

Either make a regex filter as a the_content filter,

add_filter('the_content', 'my_regex_filter', 1); // Execute as early as possible

function my_regex_filter($content) {
    $regex = get_shortcode_regex(array('showgallery'));
    $content = preg_replace("https://wordpress.stackexchange.com/".$regex.'/s', '[$1gallery$3$4$5$6]', $content );
    return $content;
}

or rewrite your add_shortcode function.

remove_shortcode('showgallery');
add_shortcode('showgallery', 'my_showgallery');

function my_showgallery($args) {
    return gallery_shortcode($args);
}

I would recommend the second, because it is cleaner and faster.

Edit:

Rewriting the shortcode for external plugins.

In your functions.php define the plugin version of your shortcode:

remove_shortcode('showgallery');
add_shortcode('showgallery', 'my_showgallery');

function my_showgallery($args) {
    return gallery_shortcode($args);
}

function my_template_gallery($args) {
    // Here put your custom gallery code…
}

Add to your header.php the following two lines:

remove_shortcode('showgallery');
add_shortcode('showgallery', 'my_template_gallery');

Now you got the rendered WP shortcode everywhere except on your templates.