We've probably done this before.
As most people here know, a number of US VP's have become President due to the President leaving office by death or resignation. In many cases, little thought or attention had been given to their adoption as running-mate. So it should not be hard to think up a POD whereby someone else is so adopted and then steps up.
1. 1840 -
https://en.wikipedia.org/wiki/1839_Whig_National_Convention#Vice_presidential_nominee "Because Harrison (born in Virginia) was considered a Northerner (as a resident of Ohio), the Whigs needed to balance the ticket with a Southerner. They also sought a Clay supporter to help unite the party. After being turned down by several potential candidates, including John J. Crittenden, John Bell, and Willie Person Mangum, the convention finally found its Southerner who had faithfully supported Clay and would accept: former Senator John Tyler.[3] Tyler was well known to the delegates, having previously been the running mate of Hugh Lawson White and Willie Person Mangum during the four-way Whig campaign of 1836.[4] He was easily nominated on the first ballot."
So WI Crittenden, Bell, or Mangum accept the nomination and then step up in 1841?
2. 1848
https://en.wikipedia.org/wiki/1848_United_States_presidential_election#Whig_Party_nomination
"After Daniel Webster turned down the vice presidential candidacy, Millard Fillmore received the party's vice-presidential nomination on the second ballot after defeating Abbott Lawrence, a Massachusetts politician whose mild opposition to slavery led him to be dubbed a "Cotton Whig".
So WI Webster accepts the nomination and then steps up in 1850?
3. 1864 - a biggie
https://en.wikipedia.org/wiki/1864_National_Union_National_Convention#Vice_Presidential_nomination - it looks as if Andrew Johnson was the preferred choice from an early stage.
4. 1880 -
https://en.wikipedia.org/wiki/1880_United_States_presidential_election#Republicans
"To placate the Grant faction, Garfield's Ohio supporters suggested Levi P. Morton for vice president. Morton declined, based on Conkling's advice. They next offered the nomination to Chester A. Arthur, another New York Stalwart. Conkling also advised him to decline, but he accepted."
So WI Morton accepts the nomination and then steps up in 1881?
5.
https://en.wikipedia.org/wiki/1900_United_States_presidential_election#Republican_Party_nomination
" Thomas C. Platt, the "boss" of the New York State Republican Party, did not like Theodore Roosevelt, New York's popular governor, even though he was a fellow Republican. Roosevelt's efforts to reform New York politics – including Republican politics – led Platt and other state Republican leaders to pressure President McKinley to accept Roosevelt as his new vice presidential candidate, thus filling the spot left open when Vice President Garret Hobart died in 1899. By electing Roosevelt vice president, Platt would remove Roosevelt from New York state politics. Although Roosevelt was reluctant to accept the nomination for vice president, which he regarded as a relatively trivial and powerless office, his great popularity among most Republican delegates led McKinley to pick him as his new running mate."
https://en.wikipedia.org/wiki/1900_Republican_National_Convention#Vice_Presidential_nomination
"Entering the convention, many had expected that the ticket would consist of President McKinley and New York Governor Theodore Roosevelt.[5] However, Ohio Senator Mark Hanna maneuvered to keep Roosevelt off the ballot, instead proposing Navy Secretary John D. Long of Massachusetts or Iowa Representative Jonathan P. Dolliver.[5] Without the support of McKinley, Hanna's efforts fell short.[6] Roosevelt himself did not particularly want to abandon his position of governor, but he desired to run for president in 1904 and when the party nominated him, he accepted the position.[6] Roosevelt's nomination was spearheaded by bosses Matthew Quay of Pennsylvania and Thomas C. Platt of New York, the latter of whom wished to find a different job for the reformist Roosevelt.[6]"
So WI Roosevelt flatly refuses the nomination? Does Long or Dolliver get nominated and then step up in 1901?
6.
https://en.wikipedia.org/wiki/1920_United_States_presidential_election#Republican_Party_nomination
"Once the presidential nomination was finally settled, the party bosses and Sen. Harding recommended Wisconsin Sen. Irvine Lenroot to the delegates for the second spot, but the delegates revolted and nominated Coolidge, who was very popular over his handling of the Boston Police Strike from the year before. The Tally:
Vice Presidential Balloting,
Republican Nat'l Convention 1920
Calvin Coolidge 674.5
Irvine Lenroot 146.5
..."
https://en.wikipedia.org/wiki/1920_Republican_National_Convention#Vice_Presidential_nomination
"Before Harding was nominated, [California Senator Hiram] Johnson, Kansas Governor Henry Justin Allen, Massachusetts Governor Calvin Coolidge, Wisconsin Senator Irvine Lenroot, Kentucky Governor Edwin P. Morrow, and Harding himself were all seen as possible vice presidential nominees.[8] Once the presidential nomination was finally settled, Harding and the party bosses asked Johnson to join the ticket as a progressive balance to Harding.[13] When Johnson turned down the offer, they approached Lenroot, who accepted.[13] However, when Illinois Senator Medill McCormick stood up to nominate Lenroot, several delegates began to shout for Coolidge.[13] Though initially he had only been nominated after the refusal of Senator Lodge (another Massachusetts man), a groundswell of support built up for Coolidge, who won the nomination over Lenroot; historian Donald R. McCoy called it the first time since the 1880 nomination of James Garfield that "the delegates had taken control of a Republican convention".[14] Coolidge, who was not at the convention during the vice presidential nomination, agreed to join the ticket."
So WI Johnson or Lodge is prepare to let his name go forward, is nominated, and steps up in 1923?
7. 1944 - a biggie
https://en.wikipedia.org/wiki/1944_United_States_presidential_election#Democratic_Party_nomination
"Numerous moderate party leaders privately sent word to Roosevelt that they would fight Wallace's re-nomination as vice president and proposed instead Senator Harry S. Truman, a moderate from Missouri. Truman was highly visible as the chairman of a Senate wartime committee investigating fraud and inefficiency in the war program. Roosevelt, who personally liked Wallace and knew little about Truman, agreed reluctantly to accept Truman as his running mate to preserve party unity.[3] Even so, many delegates on the left refused to abandon Wallace, and they voted for him on the first ballot.[4] However, enough large Northern, Midwestern, and Southern states supported Truman to give him victory on the second ballot."
https://en.wikipedia.org/wiki/1944_Democratic_Party_vice_presidential_candidate_selection
There have been extensive previous discussions about this - and people were well aware at the time about Roosevelt's declining health.
8. 1960
https://en.wikipedia.org/wiki/1960_United_States_presidential_election#Democratic_Party -
"Robert F. Kennedy ... hated Johnson for his attacks on the Kennedy family, and ... favored labor leader Walter Reuther ... Robert Kennedy visited with labor leaders who were extremely unhappy with the choice of Johnson, and, after seeing the depth of labor opposition to Johnson, he ran messages between the hotel suites of his brother and Johnson, apparently trying to undermine the proposed ticket without John Kennedy's authorization and to get Johnson to agree to be the Democratic Party chairman, rather than vice president."
https://en.wikipedia.org/wiki/1960_Democratic_Party_vice_presidential_candidate_selection
So is it possible to imagine a TL where the Kennedys decide that Johnson could cost them labor votes and decide to go with Reuther?
9. 1973
https://en.wikipedia.org/wiki/Gerald_Ford#Vice_presidency_(1973%E2%80%931974)
"According to The New York Times, Nixon "sought advice from senior Congressional leaders about a replacement." The advice was unanimous. House Speaker Carl Albert recalled later, "We gave Nixon no choice but Ford." "
https://en.wikipedia.org/wiki/1973_United_States_vice_presidential_confirmation
"President Nixon considered selecting former Texas Governor and Treasury Secretary John Connally, New York Governor Nelson Rockefeller, and California Governor Ronald Reagan." President Reagan in 1973 is an "interesting" idea.