Fugitive Doctor

An incarnation of the Doctor, identified as such by the Judoon and the Thirteenth Doctor's sonic screwdriver, had a history of working for officials on Gallifrey. She served in a role she had not signed up for, and which she sought to abandon. At one point, she hid out on Earth using a Chameleon Arch, taking on a human identity as "Ruth Clayton".

She had her TARDIS buried near the lighthouse where she stored her true identity, and trusted Lee Clayton to protect her, under cover as Ruth's husband. Lee was also tasked with reminding her to "Follow the light and break the glass" when the time was right, as this would lead Ruth to return to the lighthouse, which she was made to recall from childhood, and finally restore her Time Lord memories. (TV: Fugitive of the Judoon)

The Thirteenth Doctor later learned from that "Ruth" was one of her past incarnations from before what she thought was her first incarnation, the memory of which had been redacted from her mind. (TV: The Timeless Children)

Working for Gallifrey
For a period of time, having been consigned or recruited, the Doctor worked for the enforcers of Gallifrey. She tried to flee from them on at least one occasion, but learned that escape would not prove an easy task after she failed to get away. While working for the Time Lords, the Doctor became acquainted with Gat.

After she successfully escaped from Gallifrey, stealing a gun from Gat in the process, the Doctor fled to Earth in her TARDIS and used a Chameleon Arch to disguise herself as a human named "Ruth Clayton", with Lee Clayton acting as her protector in case of trouble. Taking up residence in Gloucester with Lee posing as her husband, Ruth Clayton made a living as a self-employed tour guide. She believed herself to be a 43-year-old human, and remembered being brought up by her parents in a disused lighthouse. (TV: Fugitive of the Judoon)

As Ruth Clayton
According to her constructed memories, Ruth moved away from home to Gloucester in mid-December 1999, seeking a life of her own. Her parents died after this point, and Ruth inherited the lighthouse from them, though she did not wish to live there. (TV: Fugitive of the Judoon)

By 2020, (TV: Can You Hear Me?) she was a regular customer at a coffee shop where she had become friendly with the barista, Allan. Allan tried to convince Ruth that Lee was not good enough for her, even keeping a dossier which recorded his suspicions regarding him. She also made a friend in elderly resident Marcia. (TV: Fugitive of the Judoon)

Hunted by the Judoon
On Ruth's supposed 44th birthday, the Judoon, under contract with Gallifrey, came to Gloucester to retrieve the Doctor, having pinned down her location due to Lee keeping hold of his old service medal. When another incarnation of the Doctor, the Thirteenth Doctor, haggled a brief reprise from the Judoon to investage matters herself, Lee distracted the Judoon to allow Ruth to escape with the Doctor. He was ultimately executed by Gat, but not before sending Ruth a cryptic text message that briefly reawakened her Time Lord persona to defend Ruth from the Judoon. In an act beyond self-defence, Ruth pulled off Captain Pol-Kon-Don's horn, escalating the Judoons' hostility and causing the Thirteenth Doctor to inquire after her true identity.

As memories began surfacing unexpectedly, she and the Thirteenth Doctor then travelled to the lighthouse, where Ruth followed her imperative to "break the glass". After shattering the biodata module with the push of a button, Ruth's Time Lord personality was reactivated and she became "the Doctor" again. (TV: Fugitive of the Judoon)

Memories restored
With her true self restored, the "Fugitive" Doctor retrieved the rifle she had stolen from Gat, quickly changed into her old clothes, and went outside to introduce herself to the Thirteenth Doctor, who had dug up her TARDIS. The "Fugitive" Doctor then teleported the two of them onboard, where she learned of the Thirteenth Doctor's identity. Since she didn't recognise the Thirteenth Doctor, she assumed she was encountering a future incarnation of herself, only to learn that the Thirteenth Doctor had no memory of ever being her either.

Before either Doctor could figure out what was going on, the "Fugitive" Doctor's TARDIS was captured by the Judoon. Onboard the Judoon ship, the two Doctors encountered Gat, and the "Fugitive" Doctor returned her rifle to her. She attempted to pass off her thirteenth incarnation as "nobody", but the Thirteenth Doctor ultimately revealed who she was. To the "Fugitive" Doctor's surprise, the Thirteenth Doctor also claimed that Gallifrey had been destroyed by "a madman", sharing her memories of this with Gat as proof that she was not lying.

Gat tried to execute both Doctors, despite the "Fugitive" Doctor's pleas for her to stand down, having sabotaged the rifle, and it backfired on Gat, killing her. Setting aside the Thirteenth Doctor's horror at her actions, the "Fugitive" Doctor retrieved the firearm and reversed her alterations. Using the weapon, she threatened the Judoon with it, pointing out that, now they were in interstellar space, crime was not defined, and they had no jurisdiction. After the Judoon let them leave, the "Fugitive" Doctor returned her thirteenth incarnation to Earth, noting that they couldn't both be right about each other's place in their chronology. (TV: Fugitive of the Judoon)

Matrix projection
After being trapped in the Matrix by, the Thirteenth Doctor encountered a projection of this Doctor who she speculated was possibly the Matrix playing more games with her. The "Fugitive" Doctor suggested that the Doctor had summoned her to help and the Doctor questioned where the "Fugitive" Doctor fit into her life and if all of her memories of this incarnation were erased. The "Fugitive" Doctor stated that she didn't have the answers to the Doctor's questions and asked if they would even help if she did have them. The "Fugitive" Doctor pointed out that the Doctor was never limited by who she was before and stated that the Doctor had no time to be tired as the universe needed the Doctor too much. After the Doctor stated that she didn't know how to defeat the Master, the "Fugitive" Doctor told the Doctor that she did know and reminded the Doctor of the one thing "he" said that has been nagging at her. The "Fugitive" Doctor suggested that the Doctor should "blow the mind" of the Matrix before disappearing. (TV: The Timeless Children)

Personality
The "Fugitive" Doctor had a low opinion on the Thirteenth Doctor's sonic screwdriver, terming it a "gizmo", and arrogantly claimed she was too smart to use one for herself.

She exhibited a more violent persona than many of her other incarnations, threatening to kill a Judoon platoon, and arming herself with a gun as well as programming Gat's gun to blow its owner up.

She also showed dislike for the Thirteenth Doctor. After instructing her to keep quiet and let her do all the talking, the "Fugitive" Doctor was furious with her for interfering in her plans for Gat. (TV: Fugitive of the Judoon)

Habits and quirks
The "Fugitive" Doctor spoke in a traditional London accent, both while under the impression of being Ruth Clayton, and as her true self.

In a manner similar to the Thirteenth Doctor, she gives out points as people got something right. (TV: Fugitive of the Judoon)

Skills
The "Fugitive" Doctor had the combat skills to overpower a platoon of Judoon with her bare hands, and even had the strength to remove Captain Pol-Kon-Don's horn with one hand.

She could drive a car. (TV: Fugitive of the Judoon)

Appearance
The "Fugitive" Doctor resembled a dark-skinned middle-aged woman. She had black hair, which she wore in dreadlocks. (TV: Fugitive of the Judoon)

Clothing
The "Fugitive" Doctor wore a royal blue tweed frockcoat with 14 silver buttons. She wore a kente shirt under a double-breasted waistcoat of navy blue moleskin with 16 bronze buttons. She also wore black trousers and boots. (TV: Fugitive of the Judoon)

Behind the scenes
While Fugitive of the Judoon left it unclear where the "Fugitive" Doctor fits into the Doctor's life, Chris Chibnall confirmed to The Mirror that, as stated in the episode, "She is definitively the Doctor." He went on to say that "There's not a sort of parallel universe going on; there's no tricks."

First POC to play the Doctor?
The "Fugitive" Doctor was the first within the DWU to be played by a non-white actor, Jo Martin. Some stories have previously shown the Doctor briefly taking over the bodies of non-white humans, and so getting played by those who were cast for those characters, and others have already indicated that non-white incarnations do in fact exist somewhere along the Doctor's timeline.

Daniel Anthony played the Eleventh Doctor in The Sarah Jane Adventures story Death of the Doctor while he occupied the body of Clyde Langer, and Damian Lynch played the Third Doctor in the audio story Ghost in the Machine while he was using the body of Benjamin Chikoto to communicate.

More concretely, Russell T Davies' Rose novelisation shows that Clive Finch has collected a photograph of, among other known incarnations, a tall, bald, black female Doctor who wields a flaming sword.