Message between users does NOT use cron.
You can visit http://example.com/wp-cron.php?doing_wp_cron to fire cron and wait for sometimes to see if you receive any email. (Change example.com with your website)
You can also install https://wordpress.org/plugins/advanced-cron-manager/ to view your cron jobs and debug.
Let me know.
Or you can give me your website admin and ftp access to debug.