We have seen students send in their paperwork years after completing it, and the RCMP still processed it. It’s impossible to give this question a blanket answer as licensing is handled on a case-by-case basis federally.
If you are in doubt, call the Canadian Firearms Program at 1-800-731-4000 and follow their instructions.
We encourage students to take a digital copy of their “proof of pass” Course Report forms, which they receive when they pass testing. In the event that a student application is lost in the mail, this copy becomes very useful in proving the course was indeed taken.