Prime Minister of the United Kingdom

The Prime Minister was the political head of the United Kingdom. Several other nations had the same office title.

20th century
The Doctor spoke of having met and known then-Prime Minister Winston Churchill during World War II. In his sixth incarnation, he encountered Churchill twice before he became Prime Minister and prevented the Players from altering the man's timeline. (PDA: Players)

He would also act as a consultant to Margaret Thatcher. (SP: Doctor Who: The Ultimate Adventure)

In the late 20th century Brigadier Lethbridge-Stewart consulted with a Prime Minister known only as "Jeremy". (DW: The Green Death) A few years later, he conferred with another, unnamed female Prime Minister. (DW: Terror of the Zygons)

On 6 May 1997, the Prime Minister, a man, is assassinated by his own bodyguard while in Washington during the Lord Xznaal crisis. (NA: The Dying Days)

21st century
In the early 21st Century, Sherilyn Harper of the far-right New Britannia party came extremely close to being voted in as Prime Minister. (BFA: The Fearmonger)

An unnamed male Prime Minister incurred the wrath of ICIS and related interests in the government due to increased links with the European Union. After he signed the Euro-Combine Treaty, this conspiracy started a wave of terrorism in the hopes of getting him to declare martial law and give ICIS the power to take charge. He later declared he would shut down ICIS over this (BFA: The Longest Night), but they briefly remained in existence for a while longer. (BFA: The Wasting)

In early 2006, following the 2005 general election, the United Kingdom has a male Prime Minister (possibly the same one). He is killed by the Slitheen, and the position of Acting Prime Minister is then assumed by Joseph Green, who is actually Jocrassa Fel-Fotch Passameer-Day Slitheen wearing the slain Green's skin. Jocrassa and his brothers are later killed by a missile strike on 10 Downing Street. (DW: Aliens of London / World War Three)

By Christmas of that year, Harriet Jones had taken over from him, though she earned the disfavour of the Doctor, who ensured her political downfall. (DW: The Christmas Invasion)

An unknown Prime Minister was in office until 2008, when "Harold Saxon" was elected. Shortly after his election, he gassed his entire cabinet and introduced the Toclafane to the human race. (DW: The Sound of Drums)

Following on from "Saxon", Aubrey Fairchild became Prime Minister (NDA: Beautiful Chaos). During the 2009 Dalek invasion, Gwen yells to Jack that they have lost contact with the Prime Minister's plane - presumably this is Fairchild, who is implied to have died.(DW: The Stolen Earth)

Alternate timelines
In the other Earth of Cybus Industries, Great Britain had the office of President rather than Prime Minister. (DW: Rise of the Cybermen / The Age of Steel)

1940s

 * Winston Churchill

1980s

 * Margaret Thatcher

2000s

 * Tony Blair (possibly pre-2006) - mentioned by Mickey Smith (DW: Rise of the Cybermen), probably Harriet Jones' predecessor whom the Slitheen killed. (DW: Aliens of London) Following the Slitheen crisis, his death was officially deemed an accident. (whoisdoctorwho.co.uk)
 * "Joseph Green" (acting 2006) - Slitheen in disguise. (DW: Aliens of London/World War Three)
 * Harriet Jones (2006-07) (DW: World War Three, The Christmas Invasion, The Stolen Earth)
 * Unknown Prime Minister (2007-08) who took over "following the downfall of Harriet Jones", and was in office for about 18 months
 * "Harold Saxon" (2008, negated timeline) (DW: The Sound of Drums/Last of the Time Lords)
 * Aubrey Fairchild (2008 - 2009) who presumably took over from Harold Saxon (NDA: Beautiful Chaos, SJA: Revenge of the Slitheen) and was apparently killed during the Dalek invasion when the plane he was travelling on was lost. (DW: The Stolen Earth)