When I check your website using Google Chrome Developer Tools I see this 403 error:
Failed to load resource: the server responded with a status of 403 (Forbidden) http://designyourownthings.uk/?wc-ajax=get_refreshed_fragments
When I click the URI|URL above I see this String loaded in the Browser Window:
{"fragments":{"div.widget_shopping_cart_content":"<div class=\"widget_shopping_cart_content\">\n\n<ul class=\"cart_list product_list_widget \">\n\n\t\n\t\t<li class=\"empty\">No products in the basket.<\/li>\n\n\t\n<\/ul><!-- end product list -->\n\n\n<\/div>"},"cart_hash":""}
The first thing that looks like a problem is this: new lines \n and tabs \t are in the String. So the question is should php trim() http://php.net/manual/en/function.trim.php be used to strip these characters out of the String? The other question is the script is not being processed and is instead being outputted directly to the Browser Window as raw code, which appears to be an encoding issue/problem.
Doing a Google search for this issue/problem I found this WooCommerce support forum topic: https://wordpress.org/support/topic/wc-ajaxget_refreshed_fragments-1
So what I think is happening is a security rule in BPS is blocking the new line characters in the String and there is another pre-existing problem with a WooCommerce script. WooCommerce files involved: /woocommerce/assets/js/frontend/cart-fragments.min.js and /woocommerce/includes/class-wc-ajax.php. I have replied to your email and requested an Administrator login to this website to investigate this further.