What's interesting, the president's term is up at noon on the 20th of January 2021. If no one has qualified as president by that time, the Speaker of the House would serve as acting president until someone has qualified as president. The article explains fully. In this case, it very well could be the current Speaker Pelosi.
Regardless of all the scenarios provided, election or no election, Trump's and Pence's terms are up at noon, 20 Jan 2021. There's no way they could continue to serve.
It's high past time that we start electing Americans to congress and the presidency who put America first instead of their political party. For way too long we have been electing Republicans and Democrats who happen to be Americans instead of Americans who happen to be Republicans and Democrats.