Let's hope we don't find out what happens if they aren't done.
What happens is laid out in Title 3 of the U.S. Code. That sets the date for the meeting of the electors. This year, that's December 14.
It says that a State's appointment of electors is conclusive if done in accordance with the State's laws as they existed at the start of the election within 6 days before the meeting of the electors. That would be December 8.
But it is rather forgiving if those deadlines are not met.
If the State has not appointed electors by that date, it may do so by December 14, but then the appointments may not be considered conclusive.
If the electors cannot (or do not) meet on the appointed date, they can do it at a later date as determined by the State's legislature.
If no certification of electors is received by the fourth Wednesday of December (December 23) the President of the Senate (the Vice President) sends a demand for the certification to the State's Secretary of State to which the Secretary must send the certificate by mail immediately. (Of course, "immediate" is somewhat subjective.)
That resolves the issues of deadlines. But there are other considerations.
The votes are counted in a joint session of Congress on January 6. During the counting, an objection to a vote can be made in writing if it is signed by one member of the House of Representatives and one member of the Senate. At that point, the counting stops. Both Houses convene to vote on the objection. The objection is only upheld if both Houses agree. There are time limits set by law for speaking prior to the vote, so there is no filibuster.
A vote cannot be rejected if it was conclusive under the laws above and was "regularly given" (a rather nebulous term).
The decisions that Congress can make in such situations are not exactly clear. For contested votes, they would probably invalid those votes. They maybe could send it back to the State.
If the votes are invalidated, there is some question on how that would affect the election. A candidate needs a majority of votes from appointed electors (270). If the votes are invalidated by Congress, does that change? Do they invalidate the votes (270 still needed) or do they invalidate the appointed electors (majority of how ever many electors there end up being).
And what if the majority of Congress is Party A and the presidential candidate with the most electoral votes is from Party B and Congress decides that they will just arbitrarily contest the votes for the Party B candidate and, with their majority, rule that those votes are not valid. That would be clearly against the law...but what then? Probably goes to the Supreme Court. Some vague Constitutional objections. But who ultimately has authority? It isn't clear. This would probably be a constitutional crises. And would probably only occur if the country is on the brink of collapse and destruction.
That is unlikely to happen.