Unfortunately I don’t have any suggestions, you probably have to check your PHP logs and even your mail server logs to see is the email being sent. I am using wp_mail() to email the backup.
You probably can try http://tareq.wedevs.com/2012/03/log-wordpress-outgoing-mails-for-debugging/
Google is your best friend. Try “wp_mail debug”
I’ve been using the automatic emailed backup for about a year. I have it set to once a week sent to my gmail.
I recently realized that the last weekly i received by email was dated April 3.
I added some debugging commands to wp-dbmanager.php but to make a long story short;
everything in wp-dbmanager, and my server is fine and apparently gmail began blocking the email for some unknown reason; same story with yahoo mail.
When i send it to my own domain’s email everything is fine.