Technically, the US Constitution doesn't say explicitly, and the question has not arisen since the 20th Amendment ( http://constitution.findlaw.com/amendment20.html ).

Between the meetings of the first Congress and before the 20th Amendment, there was no question: the terms of the president, vice president, and Congress started on March 4, so only the outgoing Congress could decide.

The original text, at http://constitution.findlaw.com/article2.html#2.1 , agrees in this respect with the 12th Amendment, at http://constitution.findlaw.com/amendment12.html :

The Electors shall meet in their respective States, and vote ... and transmit [the votes] sealed to the Seat of the Government of the United States, directed to the President of the Senate. The President of the Senate shall, in the Presence of the Senate and House of Representatives, open all the Certificates, and the Votes shall then be counted....

Since the 20th Amendment, the assumption is that the incoming Congress shall elect them. The Senate committee report (an annotation in the 20th Amendment page above) includes

[without the 20th Amendment] If it should happen that in the general election in November in presidential years no candidate for President had received a majority of all the electoral votes, the election of a President would then be thrown into the House of Representatives and the memberships of the House of Representatives called upon to elect a President would be the old Congress and not the new one just elected by the people. It might easily happen that the Members of the House of Representative, upon whom devolved the solemn duty of electing a Chief Magistrate for 4 years, had themselves been repudiated at the election that had just occurred, and the country would be confronted with the fact that a repudiated House, defeated by the people themselves at the general election, would still have the power to elect a President who would be in control of the country for the next 4 years. It is quite apparent that such a power ought not to exist, and that the people having expressed themselves at the ballot box should through the Representatives then selected, be able to select the President for the ensuing term. . . .

I think it a very clear inference that the reason that the 20th Amendment specifies that the new Congress starts on the 3d day of January, and the presidential and vice-presidential terms start on the 20th day of January, is to provide time for Congress to handle disputed elections and to elect a President and/or Vice President if needed.

But there's nothing to say it explicitly. I can imagine a repudiated Congress deciding to count the Electoral College ballots and elect the ( | vice) president, the incoming Congress deciding to do the election again, and a resulting constitutional crisis.

The 12th amendment is pretty straightforward, and is not affected by the 20th:

The person having the greatest number of votes for President, shall be the President, if such number be a majority of the whole number of Electors appointed; and if no person have such majority, then from the persons having the highest numbers not exceeding three on the list of those voted for as President, the House of Representatives shall choose immediately, by ballot, the President. But in choosing the President, the votes shall be taken by states, the representation from each state having one vote; a quorum for this purpose shall consist of a member or members from two-thirds of the states, and a majority of all the states shall be necessary to a choice.

It's immediate, per that amendment, after electoral votes are counted; and, also, it's important to note, only the House of Representatives votes for the President if the election is hung.