Hello,
Currently, it isn't possible. This is a WordPress core issue, you can take a look at this link https://core.trac.wordpress.org/ticket/44610
The oEmbed field of Meta Box uses the WordPress function wp_oembed_get() to check and save the URL that is in the list of provider URLs and the nocookie Youtube URL isn't in the list. Check the file wp-includes/class-wp-oembed.php line 53.
When the WordPress core issue is fixed, you can use the nocookie Youtube URL on your site.
.
