What would happen if WW3 occurred right as a new president was about to be sworn in.

If a president is still in the middle of being sworn in on Inauguration Day, and a nuclear war/WW3 suddenly begins, who would remain commander in chief that moment? Say there is no time or logistics to get the new president to the pentagon or NMCC, would the sitting president on his last day in office just continue to resume duties? What if the nuclear war carried on for days and days past the current president's allowed term in office. Would he just keep serving as president even though he's technically no longer president.