- You may be looking for People's Republic of Great Britain.
The origins of the British Republic stretch back to 1936 and the outpouring of grief over the assassination of Oswald Mosley. The Leader, the parallel counterpart of the Doctor, (PROSE: Timewyrm: Revelation) accused the government of orchestrating the assassination, rallying most of the British public behind his cause. When the democratic regime collapsed in 1943 the list of conspirators in the Mosley assassination had grown to include the Royal Family, who were subsequently executed along with many other members of the establishment and Great Britain was declared a Republic. (PROSE: I, Alastair)
World War II never occurred as Adolf Hitler had been killed in a military coup in 1938, (PROSE: I, Alastair) leaving Europe free to be divided in two between Great Britain and Russia. A cordial relationship between the two powers saw both sides share their most advanced technology with each other, especially military technology. The British armed forces adopted licence-made variants of the Russian Kalashnikov and SKS rifles (PROSE: The Face of the Enemy) as well as copies of Russian amphibious vehicles such as the BTR-60. (PROSE: The Schizoid Earth)
The Leader's regime discriminated against people who were not white and Christian and enacted strict cultural policies such as banning jazz music from British radio and television, (PROSE: I, Alastair) although rock and roll (PROSE: The Schizoid Earth) and country music (PROSE: The Face of the Enemy) proved too popular to repress entirely and were tolerated.
The Republic maintained its own version of UNIT called the Republican Security Forces. The RSF fought a number of alien menaces, including a group of Bannermen who had landed in Wales in 1959, as they had in the primary universe, as well as the Great Intelligence and the Yeti, with the assistance of this universe's version of the Master, still known as Koschei.
With the aid of video surveillance recordings taken when the Third Doctor arrived in their reality, the Conclave realised that the remains of Koschei's TARDIS could offer them the means to escape their dead world and take over the other universe's Earth by impersonating their counterparts. Koschei, still grieving over Ailla's death, refused to help them, so the Security Directorate forced him to give up the information through torture, using up his remaining regenerations during the interrogations. After meeting his counterpart from a parallel universe, he begged for death. His counterpart complied by disconnecting his life-support systems. (PROSE: The Face of the Enemy)