It's been 13 months since Trump was impeached and all our memories are shot. The Senate is required to hold the trial almost immediately upon receiving the articles of impeachment. But this is a special situation since the Senate isn't in session and proforma activities will happen the first day back. January 19. If the House presents those articles between now and the 19th the impeachment trial will start at 1:00 PM January 20th.
Some key dates from last time around:
December 18. House passes articles of impeachment.
January 15: House transmits articles of impeachment to the Senate
January 21: Senate votes on rules of the trial.
January 22: Opening arguments begin.
February 5: Final vote acquitting the President.
It doesn't have to take that long. It could go much quicker. As I said, it could be done in one day, if enough of the legislators, House and Senate, were so inclined. First, the House votes to change the rules. Second, the House votes to impeach. Third, the Senate votes to change the rules. Fourth, the Senate votes to convict. Done by suppertime.
But, if that doesn't happen at this stage of the game, why not?
Or they could go through the exercise of a post term impeachment trial. In your scenario, the trial starts shortly after Joe Biden takes the oath of office. I guarantee you that the majority of Americans would thing that was incredibly stupid.
So, if they can get the votes, then change the rules and make the process quick. Get him out as soon as possible.
If the Democrats can't change the rules, then they won't be able to obtain a conviction, and they will look stupid trying.