The President of Great Britain was the head of state of Great Britain on two different parallel Earths.
On the Inferno Earth, the first British President was a man usually known by his title of the "Leader". The Leader assumed office in 1943 and established a totalitarian regime that harshly repressed dissent; however he was beloved by the British public who were enraptured by the kindly image he projected in the press and on television. (PROSE: I, Alastair) The image of the Leader became such a popular and respected symbol that posters and portraits depicting him remained omnipresent for years after his death. (TV: Inferno) It has been implied that the Leader may have been an alternate Third Doctor. (PROSE: Timewyrm: Revelation) The Leader was succeeded by Gordon Lethbridge-Stewart after the latter "arranged" the Leader's death. (PROSE: Night of the Intelligence, I, Alastair) President Lethbridge-Stewart himself was later killed in a volcanic disaster and was de facto replaced by Commander Marianne Kyle. (PROSE: The Face of the Enemy)
On Pete's World, one President of Great Britain, who knew Pete Tyler, was in office in 2007. His actual power was seemingly limited, with Cybus Industries widely being considered to be more powerful than his own government. When John Lumic informed the President of his Ultimate Upgrade™ project, the President forbade it. Lumic, however, moved ahead with his plans nonetheless, disregarding the President's orders, and creating the Cybermen. He was ultimately "deleted" by Lumic's Cybermen after their raid on Jackie Tyler's birthday party, with the government being wiped out immediately after. (TV: Rise of the Cybermen, The Age of Steel) He was succeeded by Harriet Jones, who remained President into 2010. As in the primary universe, her time in office was referred to as "the Golden Age." The Tenth Doctor warned Pete Tyler to keep an eye on her, remembering her counterpart's actions with the Sycorax in his universe. (TV: Doomsday)