This is a real puzzler, so we asked our friend @atomicanalyst for his thoughts. Here's what he said...(thread) https://twitter.com/RobotWrangler/status/1335963908960448515

There are actually at least three identical briefcases, for the President, Vice President, and the designated survivor (the last is kept in the White House Military Office and assigned to the "survivor" for events such as inaugurations and State of the Union addresses).

The incoming president (& presumably VP) receive a classified briefing on the "football" & their role in its use typically the morning of inauguration or the day before. They may receive their "biscuits" then, though the codes are not valid until they officially assume office.

At the inauguration itself, as the new president takes his oath at noon, control of the briefcase discreetly passes from the former president to the new one. Dick Cheney actually described the process in an interview some years ago:

If Trump blows off the inauguration, then presumably he will have an aide with the "football" with him until noon and an aide with another "football" will be at the Capitol ready to start following Biden around after he is sworn in.

In that eventuality, if there is not already a fourth identical briefcase in the White House Military Office, one will presumably be prepared before the inauguration.