Yehudayona……….
The President assumes office at noon on January 20th (or a VP becomes President upon the vacancy of the office ala LBJ). The oath taking need not take place immediately, it is tradition to have taken place on March 3 until the date was changed by Constitutional amendment to January 20. A new President could assume all powers at noon and have his oath administered .
I don’t know that there is a halachic problem with being sworn in on Shabbos, trhe President doesn’t have to sign anything, and doesn’t have to speak into a microphone. These are added trappings, not requirements.