Final answer:
The nurse manager's priority responsibility is to ensure that staff have completed the required safety education as it is critical for providing safe patient care and meeting compliance standards.
Step-by-step explanation:
The priority responsibility of the nurse manager should be to ensure staff have completed required safety education. This is crucial because it directly relates to maintaining a safe treatment environment and adhering to regulatory compliance standards, which is essential in healthcare settings. While developing work schedules and ordering supplies are also important tasks, the immediate safety and preparedness of the staff affect the direct care provided to patients. In this scenario, the task with the highest impact on patient care is verifying the completion of safety education.