Alec Douglas-Home- 18 October 1963 to 16 October 1964 (Season 1, First Story Unearthly Child aka Unearthly Child and Last story Prisoners of Conciergerie aka The Reign of Terror Part 6)

Harold Wilson- 16 October 1964 to 19 June 1970 (Season 1 to 6, First story Prisoners of Conciergerie aka The Reign of Terror Part 6 and last story Spearhead from Space Part 3)

Edward Heath- 19 June 1970 to 4 March 1974 (Season 7 (From story Spearhead from Space Part 3 onwards) to half of first half of S12 (Last story, The Sontaran Experiment part 2)

Harold Wilson- 4 March 1974 to 5 April 1976 (second half S12 to first half of s13) First story, The Sontaran Experiment part 2 and Last story Genesis of the Daleks Part Five

James Callaghan- 5 April 1976 to 4 May 1979 (Genesis of the Daleks Part Five to S16) First story Genesis of the Daleks Part Five to The Armageddon Factor Part 6

Margaret Thatcher- 4 May 1979 to 28 November 1990 (S17 to S26) First story Destiny of the Daleks Part 1 and Last Story Survival Part 3

John Major- 28 November 1990 to 2 May 1997 (Dimensions in Time to Doctor Who TV Movie) First Story Dimensions in Time and Last Story Doctor TV Movie

Tony Blair- 2 May 1997 to 27 June 2007 ( Series 1 to the near end of Series 3) First Story Rose and Last Story The Sound of Drums

Gordon Brown- 27 June 2007 to 11 May 2010 (near end of Series 3 to first half of Series 5) First Story The Last of The Timelords and Last Story The Vampires of Venice 

David Cameron- 11 May 2010 to 13 July 2016 (first half of Series 5 to Series 9) First story Amy's Choice Last story Husbands of River Song

Theresa May- 13 July 2016 to 24 July 2019 (Series 9 to Series 11) First story Doctor Mystrio and Last Story Resolution

Boris Johnson- 24 July 2019 to 6 September 2022 (S12 to Legend of the Sea Devils) First Story Spyfall part 1 Last Story Legend of the Sea Devils

Liz Truss- 6 September 2022 to 25 October 2022 (Legend of the Sea Devils to The Power of the Doctor) First Story Legend of the Sea Devils and Last Story The Power of the Doctor

Rishi Sunak- 25 October 2022 to 5 July 2024 (The Power of the Doctor to Series 14) First Story The Power of the Doctor and Last Story The Empire of Death

Kier Starmer- 5 July 2024 to 20 July 2026 (Series 15) First Story Joy to The World Last Story The Reality War 

Andy Burnham- 20th of July to Now (current)- First Story- N/A (Well unless you really want to count the Commonwealth Games Intro which had the 7th Doctor)



  

Doctor Who PM runs (Season, Series and first and Last Story) Updated (tried my best to be accurate)