Don't know enough BritPol to know how horrified I should be, but this is what I imagine the U.S. equivalent to be (with apologies to @Ulysses Orbis, who had a similar concept but went Full-On Dystopia instead of OTL But Kind Of Shit).
1933-1945: William H. Murray (D-OK)
1945-1953: James Farley (D-NY)
1953-1961: George S. Patton (I/R-CA)
1961-1963: Foster Furcolo (D-MA)
1963-1969: Price Daniel (D-TX)
1969-1973: William E. Jenner (R-IN)
1973-1977: John Jacob Rhodes (R-AZ)
1977-1981: George Wallace (D-AL)
1981-1989: John Wayne (R-CA)
1989-1993: Alexander Haig (R-PA)
1993-2001: Edwin Edwards (D-LA)
2001-2009: Brian F. Haig (R-KY)
2009-2017: Jesse Jackson, Jr. (D-IL)
2017-0000: Rudy Guiliani (R-NY)