Want some real confusion? Kill the President-elect and Vice President-elect at 11:55, and then start the attack at 12:05.
At noon, the Speaker of the House would have become President.
But what would really happen? No one knows, or will know until it happens. In theory, at noon, the new President takes over, and the transition is instantaneous, and everyone follows orders from whoever happens to be President at the time. If the incoming and outgoing Presidents disagree, then in theory at 11:59 the outgoing President issues a command, which is obeyed without question, and at 12:01 the incoming President issues a contradictory command, which is obeyed without question.
In practice? Who knows?
The only marginally similar case that I know of that was in any way like this occurred at the 1981 inauguration. It wasn't very similar, but there was a tiny bit of similarity. American hostages were being held by Iran. Jimmy Carter, the outgoing President, was negotiating for their release. The negotiations continued overnight into the morning of January 20, and military forces were on standby awaiting orders the whole time. The staff of the incoming President was being kept informed. The actual military commanders didn't transition, so the people in charge of scrambling aircraft if necessary were the same before noon and after noon. However, the commander in chief changed promptly at noon, and I'm fairly certain that if something very, very, important happened and the President had to be notified immediately, someone would have been keeping track of a watch to know who to go to.