Rewrite attachment URLs from a local version of WordPress to a remote version.

attachment_rewrite.php

// remove username:password if you don't need htpasswd auth

add_filter('wp_get_attachment_url', function($url) {

	$upload_dir = wp_upload_dir();
	$file_path = str_replace(site_url() . '/wp-content/uploads', $upload_dir['basedir'], $url); 

	// check to see if the file exists locally first
	if ( file_exists($file_path) ) {
		return $url;
	}

	return str_replace(site_url(), 'http://username:password@staging.domain.co.uk', $url);

});

Possible improvements, check if file exists locally... if not, try and load remote version.