The state department, and anything they may or may not post on their website, has no effect on the constitutionally outlined time requirement for inauguration of a new president/time the previous presidency will officially end. According to the constitution the transition must happen at noon on January 20th. Even if the new president were not to take the prescribed oath of office, the previous presidency would still end at noon on the 20th, and the new presidency will start. Without the oath the new president just cannot use any official powers of the office of the president, yet the new person would still be president.