The President of Great Britain was the leader of Great Britain on two different parallel Earths.
On the Inferno Earth two known British presidents were Gordon Lethbridge Stewart (PROSE: Night of the Intelligence) and an alternate Third Doctor (PROSE: Timewyrm: Revelation), both led authoritarian regimes that harshly repressed dissent. (TV: Inferno)
On Pete's World one President of Great Britain, who knew Pete Tyler, was in office in 2007. Following his deletion by John Lumic's Cybermen that year, (TV: Rise of the Cybermen) he was succeeded by Harriet Jones, who remained President into 2010. (TV: Doomsday)