Thomas M. answered 12/07/21
Experienced and Committed History, Social Studies, Econ Tutor
He or she is still the President-elect. This question does not specify whether the President-elect is charged with a state or federal crime, so this answer assumes that the President-elect has been indicted for a federal crime. The FBI has a convention that a serving President in general cannot be prosecuted during his or her term. Since there is no precedent for this situation it isn't possible to definitively predict what might happen, but if the President-elect refuses voluntarily to decline to serve, two possible scenarios could play out.
The President-elect might be indicted and tried for the crime and sentenced before taking office, but in that case, he or she is still not legally disqualified from taking office. Considering that trials generally take a long time, however, it is unlikely that the trial would be over before inauguration day. Once the President is inaugurated, it is most likely that the charges would be dropped, or the case postponed. The prospect of a convicted President either governing from prison, or being "sprung" to serve as the Chief Executive would be extremely damaging to the credibility of the Administration as a whole.
The second scenario might be that prosecution of the President-elect's offense might be postponed until after their term in office. In the second case, if the offense was Federal, there is a strong likelihood that the new administration would drop the charges.
In any case, Congress would have the option of impeaching and removing the President once in office. This seems to be the most likely outcome, assuming that a majority of the House of Representatives vote to impeach, and 2/3 of the Senate vote to find the President guilty.