+1 on checking cron running and another +1 for running cron three or four times in a row to get the backups to kick in. Might have to set the time just a few minutes in the future, force the running of cron via browser, a few times. Just after the time you've set, running the cron, one should see backups kick in. After seeing the running of cron and backups, change the time to run back to something reasonable/sensible. Also check your mail server logs right after.
Plus ...
Even if one has setup SMTP server correctly and the SMTP server is set up to accept mail from the Moodle server (whitelist or whatever), the Moodle server (not the Moodle software - the server OS) needs an MTA (mail transfer agent), ie something that actually sends the mail.
Check mail server logs to see if the MTA on your server is at fault for not sending the mail notifications.
See your server is setup to use CloudFlare CDN. Maybe GMail is blocking. How does that setup affect the delivery of Mail that ID's itself with the following type of header (server name changed in the example below and the example taken from a Linux server running Sendmail as the MTA):
Return-Path: <noreply@somemoodle.someserver.net>
Received: from somemoodle.someserver.net (localhost [127.0.0.1])
by somemoodle.someserver.net (8.14.4/8.14.4) with ESMTP id r41653Bv014199
for <ktask@somemoodle.someserver.net>; Wed, 1 May 2013 01:05:04 -0500
Received: (from apache@localhost)
by somemoodle.someserver.net (8.14.4/8.14.4/Submit) id r416533T014198;
Wed, 1 May 2013 01:05:03 -0500
X-Authentication-Warning: somemoodle.someserver.net: apache set sender to
noreply@somemoodle.someserver.net using -f
To: "Admin User"<ktask@somemoodle.someserver.net>
Subject: Moodle: Automated backup status
X-PHP-Originating-Script: 48:class.phpmailer.php
The Moodle server operating system mail server, BTW, doesn't need to be set up to accept mail from the outside. It does, however, have to have the ability to send.
See by the screen shot that only the Front Page backup failed. Run that manually one time to see if there are errors reported.
And ... thought that if there had been no change to a course that affected the backups (if they ran or not and actually created a .mbz backup file).
'spirit of sharing', Ken