My first shot at an alt-president list:
1953-1953: Robert Taft/ Henry Cabot Lodge JR. (Republican)
1952: Adlai Stevenson/ John Sparkman (Democrat)
1953-1957: Henry Cabot Lodge JR./ Vacant
1957-1961: William Knowland/ Harold Stassen (Republican)
1956: W. Averell Harriman/ Estes Kefauver (Democrat)
1961-1969: Lyndon Johnson/ Hubert Humphrey (Democrat)
1960: William Knowland/ Harold Stassen (Republican)
1964: Richard Nixon/ Mark Hatfield (Republican)
1969-1973: Ronald Reagan/ Mark Hatfield (Republican)
1968: Hubert Humphrey/ James Allan (Democrat) Eugene McCarthy/ Pete McClosky (Peace and Unity)
1973-1981: Ted Kennedy/ Lloyd Bentsen (Democrat)
1972: Ronald Reagan/ Mark Hatfield (Republican)
1976: Charles Matthias/ George HW Bush(Republican)
1981-19xx: Bob Dole/ Howard Baker (Republican)
1980: Jerry Brown/ Geraldine Ferraro (Democrat) Marion Barry/ Jesse Jackson (Americans for Equality)
The basic idea is that Taft beats out Eisenhower for the nomination in '52 and manages to win the election, albeit by a smaller margin. Taft oversees the end of the Korean War on roughly the same terms as OTL, along with a rollback of some smaller New Deal programs, before dying of cancer in July 1953. Lodge, his VP, then takes over, and proceeds with a largely caretaker type administration. Notably, however, He does far less to counteract Joe McCarthy than Eisenhower. This, combined with conservatives feeling cheated by Taft's early death, leads McCarthy to primary Lodge in the 1956 election. McCarthy initially looks poised to win out, but is assassinated shortly before the convention. In his place, William Knowland is put forth as a candidate, and easily wins the convention and election.
Domestically, The Knowland administration sees the passage of the first federal civil rights act, along with the outlawing of the Communist Party and a greater rollback of New Deal programs. This trend towards rabid anti-Communism carries over to foreign affairs also, where Knowland ramps up support for French forces in Vietnam and the federal government of Cuba. This creates an even larger rift between the U.S. and the U.S.S.R.. This tension, pared with a minor economic downturn, sees the Democrats, led By Lyndon Johnson, swept into power in 1960.
The Johnson administration begins with a large focus on domestic affairs, with Johnson promising a "Great Society" built upon expanded welfare access and (with some poking and prodding from VP Humphrey) civil rights. This relative domestic bliss however, is offset by the official breakout of armed conflict in Vietnam between the Viet Cong and U.S. in 1963, which swiftly eats up Johnson's attention. Promising an end to the conflict by 1966, Johnson is re-elected in '64, only to fail to deliver on this promise. This creates rising tensions at home, as anti-war protesters team up with frustrated civil rights activists and anxious young adults to form The Peace and Unity Ticket in 1968. Backed by Anti-war Republicans and Democats, the Peace and Unity ticket siphons off enough support from the Democrats to hand the White House to Ronald Reagan that year.
The Reagan years prove to be some of the nations most challenging and polarizing. Despite bringing an end to the Vietnam War and strengthening the U.S. position against the Soviets, Reagan relied on increasingly harsh measures to quell chaos at home, leading to fierce criticism from all sides of the political spectrum. He later lost re-election to Massachusetts Governor Ted Kennedy.
From there, the timeline gets rather sketchy. I roughly imagined Kennedy as this sort of Bill Clinton-esque figure, with a moderate economic record and a tough on crime stance. The idea was that he would serve 2 successful terms before passing the torch to Jerry Brown in 1980, only to have civil rights activists split away from the party due to crime law frustration, handing the presidency back to Republicans.
(It is far far better than my first list on here back in 2011)