I got extremely carried away doing a list to explain an infobox that
@Gonzo did in his test thread on AH.com which I will link in the title to this list.
A Theoretical Look... Backward?
Presidents of the Second Republic of the United States of America
1909-1913:
William Howard Taft (Republican)
1908 (with James S. Sherman) def. William Jennings Bryan (Democratic)
1913-1917:
Theodore Roosevelt (Progressive)
1912 (with Hiram Johnson) def. Champ Clark (Democratic), William Howard Taft (Republican)
1917-1919:
Theodore Roosevelt (Second National Union)
1916 (with Thomas R. Marshall) def. Hiram Johnson ('Peace Union')
1919-1921:
Thomas R. Marshall (Second National Union)
1921-1925:
Leonard Wood (Independent)
1920 (with Herbert Hoover) def. William Gibbs McAdoo (Democratic), Hiram Johnson (Progressive), Frank Orren Lowden (Republican)
1925-1927:
Leonard Wood (Americanist Union)
1924 (with Robert L. Bullard) def. Herbert Hoover (Progressive), William Gibbs McAdoo (Democratic), Irvine Lenroot (Republican)
1927-1933:
Robert L. Bullard (Americanist Union)
1928 (with Alvin M. Owsley) def. Joe T. Robinson (Constitutional Union), Theodore G. Bilbo ('Klan Kandidate' Democrat)
1933-1938:
Hugh S. Johnson (Americanist Union)
1932 (with Ellison D. Smith) def. Herbert Hoover (Constitutional Union)
1936 (with Ellison D. Smith) def. no effective opposition
Directors of the Third Republic of the United States of America
1938-1945:
Hugh S. Johnson (Americanist Union)
1940 def. Ellison D. Smith (Independent)
1945-1953:
Huey Long (Americanist Union)
1944 def. no effective opposition
1948 def. no effective opposition
1952 def. no effective opposition
1953-1958:
Earl Long (Americanist Union - Longist Faction)
1956 def. Alvin M. Owsley (Americanist Union - Legionary Faction)
1958-1961:
Dwight D. Eisenhower (Independent leading Military Regime, backed by Legionary Faction)
Presidents of the Fourth Republic of the United States of America
1961-1965:
Dwight D. Eisenhower (Constitutional Union)
1960 (with Henry Cabot Lodge) def. Strom Thurmond (Americans United)
1965-1969:
Henry Cabot Lodge (Constitutional Union)
1964 (with George Wallace) def. Hubert Humphrey (Independent)
1969-1973:
George Wallace (National Renovation)
1968 (with Winthrop Rockefeller) def. Hubert Humphrey (Progressive)
1973-1977:
George McGovern (Progressive)
1972 (with Pete McCloskey) def. Winthrop Rockefeller (National Renovation), Hector P. Garcia (Raza Unida)
1977-1985:
Harry Byrd Jr. (National Renovation)
1976 (with Harry D. Boivin) def. Pete McCloskey (Progressive), Xenaro Ayala (Raza Unida)
1980 (with Harry D. Boivin) def. Frank Church (Progressive), Cesar Chavez (Raza Unida)
So in this world, Taft takes a more forceful position in trying to back up Diaz, drawing America into an unpleasant multi-sided guerrilla war. Clark triumphs over Wilson at the Democratic Convention and Roosevelt is able to articulate a stronger position, triumphing over the Republicans and Democrats alike. He sponsors a 'progressive alternative' in Mexico, but the war causes a big refugee crisis and distinctly blurs the border between America and Mexico with a lot migrants heading north to safer climes. Roosevelt then takes America into WW1 in 1915, when the new ultranationalist dictatorship of Mexico opposed to the fractious American-sponsored client state accepts a much earlier and undetected version of the Zimmerman Telegram. Many in America argue that the Mexican situation should be treated separately from the war in Europe but Roosevelt isn't having any of it and in 1916 forms a new National Union for the duration of the war with an aligned Democratic Vice President and he wins in a landslide. His death in 1919 only worsens the continuing fragmentation of American politics. The devotion of American resources and funds to the bloody war in Mexico means less support for the Entente, ironically leading a wheezing stalemate in Europe in 1920. America triumphs over Mexico and installs a puppet dictator, the progressive alternative long forgotten but a lot of Americans feel hard done by out of the war.
Military hero Leonard Wood triumphs over a divided field and is forced to deal with the post-war economic slump. Much blame is put on 'cheap migrant workers' from Europe and Mexico who come to America from their wartorn and depressed home nations. He forms the Americanist Union ahead of the 1924 election, which is grounded in 'common sense' economics, anti-socialism and nativism. With the Republicans increasingly irrelevant and the Democrats more and more confined to the South, they win in a landslide. Wood's death in 1927 leads to the Bullard Presidency, where the Americanist Union becomes increasingly 'fascist'. Progressives and fellow travellers in the Republican and Democratic parties gather together in Opposition. Then in 1929 the Square Mile Crash occurs, leading to rise of the Popular Front in Britain, and a renewed economic crisis in America. Bullard's common sense economics aren't enough and he is primaried by Hugh S. Johnson who has a far more radical solution to America's travails. He narrowly triumphs over Hoover in 1932, having integrated the white supremacist rump of the Democrats, and over the next four years slowly tears the Constitution to shreds. After a farcical election in 1936 where there is no actual opposition, he takes the final step and establishes America is a fascist military dictatorship.
The dictatorship lasts until the 1950s when tensions between the Long political machine and the military establishment grow too much to bear. With the death of Huey Long there is some hope for change, but his mentally unstable brother triumphs over the elderly Owsley. The dictatorship disintegrates rapidly and retired General Eisenhower launches a military coup that re-establishes a much amended Constitution and he wins the 1960 Presidential election on the old Hoover Constitutional Union ticket.
The Constitutional Union disintegrates when Lodge nominates George Wallace which is seen as shaking hands with the remnants of the Americanist Union. The work of Reconstruction argue the Progressive affiliates who break away. What remains of the Constitutionalists argue that Reconstruction isn't really necessary, whats needed is more like... Renovation. And when Wallace runs for his own term in 1968 its on the National Renovation platform. Meanwhile, the Chicano population of the Southwest ballooned under the Americanist Union, hundreds of thousands being recruited to work in the fields while America's manpower was forced into conscription and into the bloody wars of 'Anti-Socialist Defence' in Latin America, Subsaharan Africa and East Asia. With the Progressive Party being socially liberal, this large Catholic population turns instead to an emergent political movement called Raza Unida which gains in leaps and bounds over the following years. The election of McGovern is done with the votes of many Chicanos, but slow progress of his term and the increasingly socially liberal and economically centrist position of McCloskey leads to Raza Unida getting a shot in the arm. With what could have been a loyal voting block lost, Harry Byrd is confident enough in 1980 to run for a second term and wins, the first two term President since the establishment of the Fourth Republic.