One of the main reasons for having a Learning Management System (LMS) is the reporting function. Reports are used to assess e-learning uptake, Return on investment (ROI) and to justify an LMS upgrade. They are also utilised to provide evidence that a user has completed the training that has been assigned to them.
Compulsory doesn't mean 'compulsory' if there is no consequence
There is no point communicating an e-learn is compulsory to complete if the feedback loop is not acted upon. Run weekly reports, and use the 'chain of command' to communicate non-completions. If required, write a policy that clearly articulates what occurs to staff, should compulsory e-learning not be completed.
Failure to act upon non-completions results in weakening the message behind what is considered by the business to be important, mandatory learning. It also creates risk associated with failing to demonstrate that legislated training has occurred.
"There's something wrong with the system"
Depending upon the maturity of your LMS implementation, and the stability of your IT operating system, there can be times when staff have completed an e-learning module but the system has not registered completion. Under these circumstances, it is important to take a balanced view - a stand of giving the staff member the benefit of the doubt, and of checking to ensure they have understood the topic. Consider asking the staff member questions about the e-learning content. Check their understanding before closing the e-learning course from the 'back end' LMS database.