Hi Andy,
The conclusion is the gallery code is being loaded properly but the gallery javascript is not being fired. You have a lot of javascript code coming from the theme / plugins (including other lightboxes) and there must be a conflict with part of the js or one of the plugins.
I didn’t want to disable anything on your live site, but if you can do it when you have low traffic do as follows:
1. enable Justified Gallery
2. create a gallery for of the posts (make sure it has typ=”justified” parameter in the shortcode)
3. disable javascript driven plugins one by one and each time check the justified gallery display
This way you should be able to find the plugin responsible.
Hope this helps,
Bartosz