Daryl, what version of Moodle do you use? I'm using 3.1.2 and have the same problem - I don't get notified when a student has submitted an assignment. All the settings are set to get notified but it doesn't work.
The same applies for when I grade the assignment - the students don't get notified... well sometimes the email goes to them with 40-120 min delay. I worked with my hosting provider, they can't figure out what is wrong either.