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