Kenneth Clarke

A man named Clarke was a Prime Minister of the United Kingdom. According to the Eighth Doctor, he immediately succeeded Tony Blair. (PROSE: Interference - Book One)

Behind the scenes
Clarke's forename wasn't given in Interference, but the common-sense interpretation of the passage is that author Lawrence Miles was referencing the real-life Conservative politician. (However, it's also possible that the reference was to .)

In real life, Clarke contested the Conservative Party leadership in 1997, 2001, and 2005 but never became Conservative Party leader or Prime Minister.

If Clarke really did follow Blair in the DWU, he'd logically have had to have been the party leader instead of and won the 2005 election. If that's the case, then he was the Prime Minister who was murdered by the Slitheen in 2006. (TV: Aliens of London)

Clarke was played by Roland Oliver in and Kevin McNally in.