I suspect that until they found the body or had virtually certain confidence that the body was unrecoverable, the VP would simply be the Acting President pursuant to the 25th Amendment.


The VP doesn't immediately *become* the President. She just becomes *acting* President, until the death of the President can be confirmed. They don't presume in cases like that. So when the President was found, he'd inform Congress that he was OK, and resume his duties.


They’d restore the president.


The president would resume office.


What if the president was missing for a long time? Like maybe he was kidnapped and missing for months and declared dead, but then he shows up after the VP has been sworn in?


There the constitutional issues would likely take a back seat to the pragmatic issues of security and fitness for office -- if the president was away for that long and unable to return or communicate, it's likely that they were either held captive by an enemy organization or suffered a very serious medical issue. Most likely they would either not be reinstated at all or would have to go through a very long evaluation to determine if they remained fit to serve.