Four surviving elementals

The four surviving elementals were the only four Time Lords left after the end of the War in Heaven. (PROSE: The Adventuress of Henrietta Street, Trading Futures, The Gallifrey Chronicles) They ruled all of time and space near the end of the universe as the Imperial Family. (PROSE: Father Time, COMIC: Miranda)

Biology
Despite resembling ordinary people, the survivors had genetic privileges that set them apart from all other life in the universe. They were each born with two hearts, and Ferran believed that their blood was continually twisting and reforming. This birthright gave them great gifts. (PROSE: Father Time)

However, living in a universe without Gallifrey had its consequences. The second heart of a Time Lord provided a link between them and their homeplanet, but with said homeplanet no longer existing, their second hearts became poison. The Doctor's second heart pushed him to the brink of death before he was cured when Sabbath removed it and planted it within himself. (PROSE: The Adventuress of Henrietta Street) The Doctor was able to adapt to having only one heart. (PROSE: Hope) Chloe and Erasmus similarly had to remove their second hearts after they began to wither after the War. (PROSE: Timeless)

The four
A great number of sources agree that the Eighth Doctor was one of the Time Lords to survive the War. (PROSE: The Adventuress of Henrietta Street et al) Due in large part to the fact that the Matrix was downloaded into his brain, (PROSE: The Gallifrey Chronicles) the Eighth Doctor had very few memories of his life before the end of the War. He and his damaged TARDIS were deposited on Earth in the late 19th century by Compassion. (PROSE: The Ancestor Cell) After living on the planet for over a century, the Doctor regained use of his TARDIS (PROSE: Escape Velocity) and travelled through time and space with Fitz Kreiner, Anji Kapoor, (PROSE: Earthworld et al) and later Trix MacMillan. (PROSE: Timeless et al)

The Man with the Rosette was another one of the four elementals. He believed that since there were only four of his people remaining, he had to maintain "certain standards". While once a bitter enemy of the Doctor, the Man with the Rosette remained amicable with the Doctor in the post-War universe. (PROSE: The Adventuress of Henrietta Street)

Near the end of the universe, the Emperor - the greatest of the elementals - ruled the universe from the Needle, (COMIC: Miranda) which was itself another remnant of the Time Lords: Savar's TARDIS. (PROSE: The Infinity Doctors) The Emperor was the last of his kind, having outlived all of the other Time Lords. Shortly after the birth of his daughter, Miranda, the Emperor was assassinated. (PROSE: Father Time) When the Doctor saw himself in his vision of the four survivors, he was seated on an ornate throne and holding baby Miranda, suggesting that he was to become the Emperor. (PROSE: The Gallifrey Chronicles)

Though Fitz Kreiner thought skulltroopers killed Romana III and Mali, (PROSE: The Gallifrey Chronicles) by one account, Romana survived to become an old woman, living near the universe's heat death, before regenerating into a fourth incarnation. She hadn't seen the Doctor in centuries, and she described the universe as a "godless age". (PROSE: Tomb of Valdemar)

The Onihr claimed to have learned the identities of all four surviving elementals. (PROSE: Trading Futures)

Other possibilities
Iris Wildthyme encountered the Eighth Doctor at least twice in the post-War universe, once in her Jane Fonda incarnation (PROSE: Father Time) and once in her Brenda Soobie incarnation. After his encounter with Brenda, the Doctor proclaimed, "But this means... I am not alone!" To which Fitz somewhat jokingly replied, "There is... another." (PROSE: Mad Dogs and Englishmen) However, the Fonda Iris, (PROSE: The Blue Angel, Interference) the Brenda Soobie Iris, (PROSE: The Blue Angel) and even some of the later Irises (PROSE: Library Pictures, A Hundred Words from a Civil War, Michael Drake) had some involvement with the War, indicating that Iris was not a definite resident of the post-War universe. Iris also said several times that she was not a Time Lord, meaning she wouldn't qualify to be one of the four surviving elementals. (PROSE: The Blue Angel, AUDIO: The Elixir of Doom et al) Iris was once hunted down by Herr Drosselmeyr because she was one of the last of her kind. (PROSE: Another Clockwork Iris)

Romana III lived on for many centuries after her final encounter with the Doctor. As an ancient woman, she parked her TARDIS in Janua Foris and told some of its natives the story of her encounter with Valdemar. After telling the story, Romana regenerated into a young, beautiful body with dark eyes and clear ebony skin. (PROSE: Tomb of Valdemar)

After her father's assassination, Miranda was sent back in time to an earlier period of history, where she was adopted by the Eighth Doctor. As she grew up, Miranda exhibited many Time Lord attributes. (PROSE: Father Time) Miranda eventually returned to her father's era to become the new ruler of the universe. (PROSE: Father Time, COMIC: Miranda) She later died during the Eighth Doctor's encounter with the Council of Eight. (PROSE: Sometime Never...) The Doctor's vision of the four elementals included baby Miranda in his arms, suggesting that she was not one of the four. (PROSE: The Gallifrey Chronicles)

Marnal was stuck on Earth as an amnesiac during the War, and he survived it in this way, unaware of his true nature. The Eighth Doctor identified him as the fifth survivor. (PROSE: The Gallifrey Chronicles)

When she left Gallifrey after its destruction, Compassion carried with her the Gallifreyan technician Nivet. (PROSE: The Ancestor Cell) However, Compassion was significantly involved in the War even after Gallifrey's destruction, so her (and Nivet) were not definite residents of the post-War universe. (PROSE: The Book of the War, Of the City of the Saved..., Warring States)

Chloe and Erasmus were also Time Lords who survived the destruction of Gallifrey in the War. They told the Eighth Doctor that they had met many other fellow survivors. (PROSE: Timeless)

Pallant speculated that he and the rest of the Needle People were the Children of Kasterborous, a race of Gallifreyan/Human hybrids who pursued interventionist policies to create a new age of universal peace, but failed and were condemned to live until the end of time in the Librarinth. (PROSE: The Infinity Doctors)

History
According to Prefect Ferran, after the War left most of space and time broken and dead, the Imperial Family subjugated the survivors for two thousand years. They grew corrupt with power, and millions of their subjects, who they viewed as lesser species, died from their neglect, cruelty, or sport.

The Senate was powerless, and the Senators fought among themselves for scraps of power and wealth, but through secret meetings, they began to form alliances with each other, and a revolution began when the Emperor's forces murdered the insurrection leader in the street. After a short and brutal civil war, the Imperial Family was wiped out, with the exception of the Emperor's daughter, who escaped back in time and was adopted by the Eighth Doctor. (PROSE: Father Time)

Behind the scenes

 * In Lance Parkin's 2002 short story School of Doom, published in Myth Makers 12, the Master and Miranda were two of the four surviving elementals. Like the Doctor, the Master had forgotten who he was and where he came from, but he wanted the Doctor and Miranda to join him and the one other elemental to claim their birthright, gather an army on Earth, and conquer the universe to root themselves to the reality.
 * In Lance Parkin's AHistory, he specifies that he had intended the four elementals in the vision in PROSE: The Gallifrey Chronicles to be the Master, Iris Wildthyme or Romana, the Minister of Chance, and the Doctor (as his Infinity Doctors incarnation or as the Emperor). He also draws parallels between the four surviving elementals and the Four Names from PROSE: The Infinity Doctors.
 * Parkin's other Doctor Who reference book, Whoniverse: A Guide, specified that the Children of Kasterborous were the heirs to the four Time Lords who survived the War.