So the cron job sends the email saying it is running, and regardless of how it is run (cli & web), the notification page always shows that the cron script hasn't been run.
I would try and rename the cron.php script at the cli directory to see if the cron job sends and error message about not finding the php script. I would also try and install a Moodle 2.6 version. I assume you have checked and double checked that everything (minimum versions required, server settings, etc.) is fine. I would assume there aren't any other Moodle installations and so there's no way you could be mixing cron jobs/installations.
The idea of having another cron job running from another server is to increase the probability of determining where the possible issue is.
As Rick suggested, some screenshots could help. We need something more to work with.
Have you enabled Debugging? Also check the web server error logs; there could be some indication of something going wrong.