The People's Republic of Great Britain was the government of Britain in Pete's World. It was ruled by a President of Great Britain, who was killed by John Lumic's Cybermen in 2007 after he refused to fund the Ultimate Upgrade project. (TV: Rise of the Cybermen) He was succeeded by Harriet Jones, under whom the People's Republic took over the Torchwood Institute. (TV: Doomsday)
Behind the scenes
Though not given the full title "People's Republic of Great Britain" in dialogue, in Doomsday, Jake Simmonds while inside the parallel London's Torchwood Tower said, "Parallel Earth, parallel Torchwood, except we found out what the institute was doing and the People's Republic took control."