Shouldn't this be in Pre-1900?
Well just going off of how it would effect the presidency if it were post-1900 here are some of the presidents we would have ended up with.
After President McKinley is assassinated, William Jennings Bryan becomes president. Meaning Teddy Roosevelt doesn't become president at this time.
After President Harding's death, James Cox becomes president. Possibly resulting in Franklin Roosevelt never becoming president.
If FDR is president in 1945, Thomas Dewey becomes president. Harry Truman likely never becomes president. Eisenhower might not either.
After President Kennedy is assassinated, Richard Nixon becomes president in 1963. Meaning like no civil rights or voting rights acts, no Medicare, no Great Society. Vietnam escalation is still likely. But Watergate either doesn't happen or it happens sooner or something like it happens.
If Nixon is president in 1974 and resigns, either Hubert Humphrey or George McGovern becomes president. We likely get some sort of universal, single payer health care system in the U.S. by 1976 with a bigger Democratic majority after the 1974 midterms.
But going back to William Jennings Bryan becoming president in 1901, that alone will change the course of the 20th century and who does and does not eventually become president in the 20th century. But it likely changes the course of history going all the way back to the death of William Henry Harrison in 1841, making Martin Van Buren president again.