- You may be looking for People's Republic of Great Britain.
The country developed a close relationship with White Russia and worked with Russia to ensure peace in Europe by crushing Nazi Germany between two vast spheres of influence. The British Republic's ties with Russia lasted well into the 1970s. Kalashnikov and SKS rifles appear to have been the main weapons of the British armed forces. (PROSE: The Face of the Enemy)
At the time of the devastation, the Doctor's counterpart was the President of the Republic, known as the Leader. He was killed when the fallout from the disaster at Eastchester destroyed London. (PROSE: Timewyrm: Revelation)
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.
Koschei's TARDIS was ruined and its central console was cannibalised by agents of the Republic, allowing them a means of travelling between parallel universes. The Republican Security Forces killed his colleague and fellow Time Lord Ailla and British Intelligence enslaved the badly injured Koschei, imprisoning him and wiring him to their computers to forcibly extract information.
Using recordings taken when the Third Doctor arrived in their reality, the RSF realised Koschei's TARDIS could help save them by sending them into the other universe, but they needed to know how. He was tortured to death several times during his interrogation, using up his remaining regenerations. 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)