Although this might not apply to judges, I remember reading that LBJ was president the instant JFK was confirmed dead. The swearing in was his first duty as president, but was not what made him president. And we all know that there is only one penalty if the president does not carry out a specified duty.
I also recall reading that Calvin Coolidge was at home in NH when Harding and the oath of office was administered by his father a local justice of the peace. I think he was later sworn again. But it all didn’t really matter.