What action should be taken by the pilot if crossing the ILS final approach course becomes imminent without an issued approach clearance?

Maintaining the last assigned heading and querying Air Traffic Control (ATC) is the appropriate action in this scenario because it prioritizes safety and adherence to air traffic regulations. If a pilot finds themselves about to cross the final approach course without an issued approach clearance, continuing with the approach could create a dangerous situation, potentially leading to a collision with other aircraft or a breach of controlled airspace protocols.

By maintaining the last assigned heading, the pilot ensures they do not inadvertently enter the final approach path for the ILS without the necessary clearance, which is critical for both their own safety and the safety of other aircraft. Querying ATC allows the pilot to receive clarification on their status and obtain instructions, ensuring they remain in communication and do not take unilateral actions that could disrupt the flow of traffic in the area.

This approach aligns with standard operating procedures, emphasizing the importance of clear communication and compliance with air traffic instructions, thereby maximizing safety during flight operations.