Here is what I think would have happened...
I think Santos would have been a one-term President, simply based on historic precedence. John Lyman would have been his Chief of Staff, while Sam Seaborn would have been the much more popular Deputy Chief of Staff. Unlike the Bartlet Administration, the Santos Administration has to deal with crisis after crisis, and most of the legislative agenda is put aside for the duration. The 2022 midterms are a complete disaster for the Democrats, who barely retain a majority. The only advantage is that Sam barely wins a California Senate seat. After four seasons, Santos is defeated by a popular moderate Republican similar to Vinick, but much younger.
The new show introduces a new White House staff, including Joe Quincy and Ainsley Hayes and other Republicans we had been following throughout the series. To keep on the Democrats we had been following for eleven seasons, the show also begins to regularly follow the staff of Senator Sam Seaborn, including Josh Lyman and Donna Moss. Other characters such as Charlie Young are re-introduced after leaving for several seasons, and Jed Bartlet still makes the occasional appearance. The 2014 election would be a historic one. Since the show is now covering the Senate, a popular Democratic Senator (Not Sam) who is a colleague of Sam runs for President in Season 15. The election is close, but the Republican wins a victory.
This opens the path for Sam to run in 2018. In Season 19, Sam runs for President and wins a victory similar to President Obama's 2008 victory. The election pits us against the popular Republican Vice President, who is kind of similar to Bob Russell but with an 'R' next to his name. It is a popular election, but the result is a foregone conclusion. Sam's Presidency is based on the Kennedy Presidency. Sam and his young wife come from the Senate to the White House, with a sort of Camelot view of things. The start of the "Second Cold War" brings new opportunities for the show's foreign policy.
Sam in Season 23 runs for re-election, and for his final term the show goes all-out. The 2026 election pits a very popular and well-liked Republican against a very popular and well-liked Democrat. The election in Season 27 is marked as the final season, and in an upset, the Republican wins. The show ends with Sam and Josh having one last meeting, the two walking to the Rose Garden for Sam's farewell address. The show fades to black.