Inferno universe

Accessed via a power accident at the Inferno Project, the Third Doctor discovered a parallel universe where Great Britain was a republic governed by a fascist regime. Although some details were known about the history of this world, the Third Doctor personally experienced it only in the environs of its version of the Inferno Project. The Third Doctor described it as a "parallel space-time continuum" (TV: Inferno) whereas the Seventh Doctor called it an alternate timeline. (PROSE: Timewyrm: Revelation)

Creation
The Sixth Doctor believed the Inferno universe had only diverged from the primary reality about fifty years before he entered it, circa the 1970s, (PROSE: The Quantum Archangel) although no incarnation of the Doctor was present when Marianne Kyle confirmed that the first known divergence was in the 19th century. When his conversation with his third incarnation's persona revealed the identity of this world's dictator, the Seventh Doctor wondered if the Timewyrm had any part in creating this timeline as part of her efforts to weaken him. (PROSE: Timewyrm: Revelation) Another account speculated that the Inferno universe was the original timeline and the prime universe was a divergent timeline created by the Great Intelligence. (PROSE: Night of the Intelligence)

19th century
The Confederate States of America won the American Civil War during the 1860s. In the war's aftermath, the American Confederation was established. (PROSE: The Face of the Enemy)

20th century
Joseph Stalin was the leader of White Russia (PROSE: The Face of the Enemy) and few Bolsheviks survived the Russian Civil War. (PROSE: The Schizoid Earth)

Professor Edward Travers died in Tibet in 1935, before ever seeing the Yeti. John Mackay survived the trip and later worked for the British Army. (PROSE: The Schizoid Earth)

The Second World War never happened since Europe was divided into allied British and Russian spheres of influence, which were able to intimidate Adolf Hitler into backing down from his sabre-rattling. Meanwhile in Asia, the Japanese successfully built their Asian Co-Prosperity Sphere. (PROSE: The Face of the Enemy) As a result of no war occurring, no equivalent of the Wellington bomber existed in this timeline and biplanes remained the standard fighters of the Republican Air Force into the early 1940s. (PROSE: The Schizoid Earth)

The British Royal Family were executed for undisclosed reasons at an unknown date prior to 1943. Brigade Leader Alastair Lethbridge-Stewart spoke about their deaths with relish. (TV: Inferno) After the execution of the Royal Family, successive presidents drifted toward fascist ideology. (PROSE: The Schizoid Earth)

In the 1950s, British military forces were allowed to establish bases in Germany and had weapon trades with Russia. The government was increasingly draconian and paranoid about spies coming in from America, Russia, or the Co-Prosperity Sphere - which was referred to as "the Chinese" in 1959. No equivalent of the Korean War had happened. (PROSE: The Schizoid Earth) This world's James Lethbridge-Stewart would later suggest that it was the lack of a Second World War against Nazi Germany that had caused this: the British people had no recent warning of what could happen if they started giving up their freedoms and had no need or desire to fight for them. (PROSE: Night of the Intelligence)

The Republic was surprised when the Alistair Gordon Lethbridge-Stewart of the regular Earth's 1969 accidentally ended up in their 1959. They suspected him of being a spy and interrogated him for four months. By the time he'd left, they'd become aware of a third alternate Earth using their world as a staging ground to send warnings through time and the Republican Security Forces was proposed to deal with scientific threats. (PROSE: The Schizoid Earth)

Shortly after its founding, the RSF fought the Bannermen in Wales, where they reverse-engineered the alien technology. (PROSE: The Face of the Enemy)

In the increasingly fascistic Republic, Elizabeth Shaw chose to join the Army instead of pursuing a scientific career after she left university and John Benton was trained to be harsh and sadistic. (TV: Inferno)

Gordon Lethbridge-Stewart had become President by the late 1960s, disposing of even the pretence of liberal democracy. He even resorted to forcibly drugging the young people who had joined a state youth organisation called the Republic Youth Army in order to impose his will on them. James Lethbridge-Stewart was alienated by his father's thirst for power and used Axos material to get himself and his family out of this reality. (PROSE: Night of the Intelligence)

The Great Intelligence launched an attack on Britain in this universe as well. Koschei, here having not become the Master, saved the world from the Intelligence but was then captured, tortured, and pumped for information for years by the Security Forces. (PROSE: The Face of the Enemy)

At some point, this universe's Doctor arrived on Earth. He took over the Republic of Great Britain and became its Leader. (PROSE: Timewyrm: Revelation) He may have been responsible for some of the technological advances. (PROSE: The Face of the Enemy) Propaganda images of the Doctor abounded, including on posters with the slogan "UNITY IS STRENGTH". (TV: Inferno)

Near the end of the alternate Doctor's rule, in the 1970s, the world was effectively ruled by the Conclave, a parallel of the United Nations Security Council, made up of the five main continents. The Conclave were based at Copernicus moon base to protect the delegates from attack and to intimidate subversive groups and enemy nations such as Australia. The reasoning was since the Moon could be seen from all parts of the civilised world, it would serve as a reminder that the Conclave was watching. The powers were Russia, India, the American Confederation, the Asian Co-Prosperity Sphere, and Britain (representing Europe). (PROSE: The Face of the Enemy) A parallel of UNIT was known as the European Axis Military Corps. (PROSE: Still Lives)

In 1970, the RSF were a parallels of contemporary British UNIT members: Brigade Leader Lethbridge-Stewart, Platoon Under Leader John Benton, and Section Leader Elizabeth Shaw, the latter having entered the military in this world. Eric Stahlman was carrying out a more advanced version of the Inferno Project. Australian Greg Sutton was a political prisoner in Britain, forced to assist Stahlman's project. (TV: Inferno)

Shortly after the Third Doctor's unplanned arrival in this universe, an accident at Stahlman's project devastated the entire surface of the planet within thirty-nine hours. (TV: Inferno) The only survivors were those in bunkers, space stations and on the Moon. This world's Doctor was killed when the fallout from the disaster at Eastchester destroyed London. (PROSE: The Face of the Enemy)

The Doctor had recognised the Leader's face was one of the regenerations he could have chosen at the trial. He would be haunted afterwards by the knowledge of what his alternate self had done. (PROSE: Timewyrm: Revelation)

While the Doctor and Jo Grant were visiting Peladon, the surviving leaders of this Earth started to cross over to the Doctor's world. They planned to kill and replace their counterparts, but UNIT (reluctantly working with thwarted the plot. The Master allowed Marianne Kyle to escape, hoping she would cause trouble for the Doctor some day. (PROSE: The Face of the Enemy)

Behind the scenes
The world is never actually referred to as "Inferno Earth" in any known DWU story, though the phrase "the Inferno universe" occurs once in The Quantum Archangel. However "Inferno Earth" serves as a useful shorthand. It's a logical in-universe term, since the Doctor accessed that particular Earth at the Inferno Project — something the novelisation tells us is colloquially known as "the Inferno".