Inferno universe

On an Earth accessed via a power accident at the Inferno Project, the Third Doctor discovered 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.

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)

The British Royal Family had been executed for undisclosed reasons at some point prior to the 1940s. (TV: Inferno) The Japanese successfully built their Asian Co-Prosperity Sphere. 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.

This world was effectively ruled by the Conclave, a parallel of the United Nations Security Council. 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. (PROSE: The Face of the Enemy)

The parallel of UNIT was known as the European Axis Military Corps. (PROSE: Still Lives) and the British contingent of this organisation was known as the Republican Security Forces. Many of the members of the RSF were the parallels of British UNIT members, including Alistair Gordon Lethbridge-Stewart, Elizabeth Shaw and John Benton. (TV: Inferno) Shortly after the Doctor's unplanned arrival in this universe, an accident at Stahlman's project devastated the entire surface of the planet within thirty-nine hours. The only survivors were those in bunkers, space stations, or on the Moon. (PROSE: The Face of the Enemy)

At the time of the devastation, the Doctor's counterpart was the President of the Republic of Great Britain, known as the Leader. He was killed when the fallout from the disaster at Eastchester destroyed London. (PROSE: Timewyrm: Revelation)

Other information

 * When his conversation with his third incarnation's persona revealed the identity of the dictator of the Inferno Earth, 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)
 * Jane Page may have been a native of this timeline as she worked for the counter-espionage agency of a dictatorial British Republic. She was removed from her timeline by Gabriel and Tanith. (PROSE: Falls the Shadow)

Behind the scenes
The world is never actually referred to as "Inferno Earth" in any known DWU story. However it 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".