It probably won't, because, in past election disputes, SCOTUS either made rulings that were "active" rulings, or made rulings that nothing could be done, citing the hard deadlines for the Electoral College. They've ruled that, barring a change in election law and/or a Constitutional Amendment, those dates are set in stone and can't be altered by a court.
The meeting of electors is set in stone. In order to change federal election law, you'd need the change to pass both houses of Congress and be signed by the President. Not going to happen. -
The electors of President and Vice President of each State shall meet and give their votes on the first Monday after the second Wednesday in December next following their appointment at such place in each State as the legislature of such State shall direct.
3 USC 7: Meeting and vote of electors
If a state tries to not send their electors, federal law requires Congress to demand their electors, and the states to comply (note that the language "shall" is mandatory, not optional language).
When no certificate of vote and list mentioned in sections 9 and 11 of this title from any State shall have been received by the President of the Senate or by the Archivist of the United States by the fourth Wednesday in December, after the meeting of the electors shall have been held, the President of the Senate or, if he be absent from the seat of government, the Archivist of the United States shall request, by the most expeditious method available, the secretary of state of the State to send up the certificate and list lodged with him by the electors of such State; and it shall be his duty upon receipt of such request immediately to transmit same by registered mail to the President of the Senate at the seat of government.
3 USC 12: Failure of certificates of electors to reach President of the Senate or Archivist of the United States; demand on State for certificate
The Congressional Research Service has been asked about this subject.
What remains clear is that only the states and Congress have the power to delay that part of the election process. “Unlike the practice of some states that allow the Governor to postpone an election during emergencies, neither the Constitution nor Congress provides any similar power to the President or other federal officials to change this date outside of Congress’s regular legislative process,” the report says.
Constitution.org: Does the Constitution allow for a delayed Presidential election?
While that is specific to actual public voting/election dates, it seems like that holds true for Electoral College dates, as well.
That was, in part, the stated reason why Florida had to stop their recount in Bush v Gore in 2000 - there wasn't enough time.
which requires that any controversy or contest that is designed to lead to a conclusive selection of electors be completed by December 12. That date is here, but there is no recount procedure in place under the state court's order that comports with minimal constitutional standards.
Justia.com: Bush v Gore
Even if he wins cases, if they can't get new, certified results as of that date, the previous ones would stand.
If the legal wrangling makes it so there is no winner that can be chosen, that doesn't mean Trump sticks around until it is resolved. His term of office ends, again, via the Constitution, and the next President gets sworn in on hard and fast dates.
The terms of the President and the Vice President shall end at noon on the 20th day of January, and the terms of Senators and Representatives at noon on the 3d day of January, of the years in which such terms would have ended if this article had not been ratified; and the terms of their successors shall then begin.
The Constitution Center: the 20th Amendment
So, no elected President or VP means that the next person in the line of succession takes over - Speaker of the House.
The next session of the new Congress starts 2 to 3 weeks (also from the 20th Amendment) before the next President takes over, so there would be a new, duly appointed and active Speaker at that time.
(a)(1) If, by reason of death, resignation, removal from office, inability, or failure to qualify, there is neither a President nor Vice President to discharge the powers and duties of the office of President, then the Speaker of the House of Representatives shall, upon his resignation as Speaker and as Representative in Congress, act as President.
3 USC 19: Vacancy in offices of both President and Vice President; officers eligible to act
So, potentially, if conservatives and Trump really, really want to drag this out, they might wind up with, for a time period, President Pelosi. My own subjective assessment is that they'd drop their legal fight unless it was looking like a slam dunk, than go down that road.