Your scenario is highly unlikely.
It takes weeks, if not months to draw up and pass a Bill of Impeachment.
For the Clinton Impeachment, the House Judiciary Committee eventually came up with four Articles of Impeachment but two were voted down, even in a Republican-controlled House.
An article charging Clinton with perjury in the civil sexual harassment suit brought against him was defeated by a vote of 229-205; and another article charging him with abuse of office by false responses to the House Judiciary Committee’s written request for factual admissions was defeated by a vote of 285-148.
Obama could easily resign in order to be pardoned by Biden while a Bill of Impeachment is being drawn up in the House. President Nixon waited until he knew the actual count of votes for removal from office in the Senate Judiciary Committee (the vote was 27-11 guilty).
How long does it take to submit a Letter of Resignation and to have a swearing in ceremony? Five minutes?
Agreed. Ousting either the president or the vice president requires 67 votes.
Hmm. If the votes were available, the crafty approach would be to remove Biden first, then impeach and remove Obama while stalling the confirmation of Biden's replacement. That would require organization and stealth. E.g., fewer than 67 GOP, so Obama thinks his Regime is safe, but some Democrat cross-overs up the sleeve. Once Biden's ousted, Obama's trapped. Well maybe not. Maybe he can pardon himself‽