https://www.law.cornell.edu/uscode/text/3/15#

I'm going to go over a few points from this code that are important to keep in mind.

Congress shall be in session on the sixth day of January succeeding every meeting of the electors. The Senate and House of Representatives shall meet in the Hall of the House of Representatives at the hour of 1 o’clock in the afternoon on that day, and the President of the Senate shall be their presiding officer.

For clarity to those who may not know, the President of the Senate is the incumbent Vice President.

(..) and the votes having been ascertained and counted according to the rules in this subchapter provided, the result of the same shall be delivered to the President of the Senate, who shall thereupon announce the state of the vote (..) and, together with a list of the votes, be entered on the Journals of the two Houses. Upon such reading of any such certificate or paper, the President of the Senate shall call for objections, if any.

Of which there no doubt will be many, from both sides of the aisle.

Every objection shall be made in writing, and shall state clearly and concisely, and without argument, the ground thereof, and shall be signed by at least one Senator and one Member of the House of Representatives before the same shall be received. When all objections so made to any vote or paper from a State shall have been received and read, the Senate shall thereupon withdraw, and such objections shall be submitted to the Senate for its decision; and the Speaker of the House of Representatives shall, in like manner, submit such objections to the House of Representatives for its decision;

Shall is commanding language. The Senate and the House must both decide upon the objections.

and no electoral vote or votes from any State which shall have been regularly given by electors whose appointment has been lawfully certified to according to section 6 of this title from which but one return has been received shall be rejected, but the two Houses concurrently may reject the vote or votes when they agree that such vote or votes have not been so regularly given by electors whose appointment has been so certified.

Regularly, in this language, means in accordance with any and all regulations, including whether or not the certification of electors was conducted in an entirely legal manner.

(..) in case there shall arise the question which of two or more of such State authorities determining what electors have been appointed, as mentioned in section 5 of this title, is the lawful tribunal of such State, the votes regularly given of those electors, and those only, of such State shall be counted whose title as electors the two Houses, acting separately, shall concurrently decide is supported by the decision of such State so authorized by its law;

If a state did not adhere to its own laws, then the electors certified by that state cannot have been lawfully certified. This applies to contesting electors as well, since they cannot be legally certified if the law was not followed to begin with either, which brings us to the next point.

(..) and in such case of more than one return or paper purporting to be a return from a State, if there shall have been no such determination of the question in the State aforesaid, then those votes, and those only, shall be counted which the two Houses shall concurrently decide were cast by lawful electors appointed in accordance with the laws of the State, unless the two Houses, acting separately, shall concurrently decide such votes not to be the lawful votes of the legally appointed electors of such State.

This allows the two houses of Congress to decide which electors to certify, in accordance with state election law, in the event that either both parties were legally certified and the Houses are in disagreement in judgement, or neither slate of electors could have been legally certified by the state.

But if the two Houses shall disagree in respect of the counting of such votes, then, and in that case, the votes of the electors whose appointment shall have been certified by the executive of the State, under the seal thereof, shall be counted.

If, and only if, no decision can be reached by Congress to settle the dueling slates, the state executive's choice shall be counted.

When the two Houses have voted, they shall immediately again meet, and the presiding officer shall then announce the decision of the questions submitted. No votes or papers from any other State shall be acted upon until the objections previously made to the votes or papers from any State shall have been finally disposed of.

This obviously includes court cases awaiting settlement by those states, since the grounds of objections cannot be disposed of if the court cases pending adjudication are included in the grounds of the objection.

TL;DR expect this to be capable of dragging out so long that it barely matters if the Supreme Court slow-walks the cases. There is no stated deadline in this code that provides a deadline for the EC votes to be acted upon except that all objections pertaining to each state have been decided upon and disposed of.