On DVD commentary, producer Phil Collinson said that Aliens of London had intended to show a Blair look-a-like as the Prime Minister's corpse. When the resemblance proved disappointing, they decided to avoid showing the body clearly. Other hints and jabs at Blair remain, but the show avoids saying outright that he's the Prime Minister: the 2005 general election had not occurred during filming, and Rose and the Doctor are unaware who won so the show can avoid being in error.
The Prime Ministers are listed in rough chronological order by their first term in office. Please note that this template only uses information from fictional sources. Real world information should not enter into this.