British monarchy

The United Kingdom operated under a constitutional monarchy.

The Doctor and the monarchy
The Doctor had numerous encounters with the Kings and Queens of Britain. So much so that Elizabeth X grew up on stories of the Doctor in the 29th century. (DW: The Beast Below)

The Doctor's relationships with the monarchs varied. The Tenth Doctor, after apparently marrying Elizabeth I (DW: The End of Time), infuriated her to the point that the next time she saw him, she ordered his execution. (DW: The Shakespeare Code) While Queen Victoria both knighted and exiled him on the same day. (DW: Tooth and Claw) However, he also had good relationships with monarchs, such as Elizabeth II. (DW: Voyage of the Damned, Planet of the Dead)

History

 * For ease of identification, the summary below bolds the names of known Monarchs.

Early history
Athelstan of England was the King of England for some time in the 10th century. He received the Cup of Athelstan in 924 as a coronation gift from Hywel, King of the Welsh. (DW: Planet of the Dead)

Harold Godwinson was King of England for most of 1066. After he defeated a Viking invasion, the Normans invaded England from the south. The Monk planned to change history by using advanced technology to aid King Harold, ensuring that his army would be better able to face the Normans. This plan was defeated by the First Doctor and Harold was defeated in the Battle of Hastings by William the Conqueror, who became the new king of England. (DW: The Time Meddler)

Richard I of England (8 September 1157 – 6 April 1199), called Richard the Lionheart, was King of England from 6 July 1189 until his death. While he led his troops in the crusades, he worried that his younger brother John of England would usurp his throne back in England. Richard assisted the First Doctor and Ian Chesterton in rescuing Barbara Wright after her capture by Arabs. (DW: The Crusade)

King John of England ruled England in the early 13th century. While the real King John was in London taking the Crusader's Oath, the android Kamelion was being used in a plot by the Master to sabotage Earth history by preventing King John's signing of the Magna Carta, an event pivotal to the development of parliamentary democracy on that planet. This plot was foiled by the intervention of the Fifth Doctor. (DW: The King's Demons)

Richard III of England (9 April 1483 - 1485) ruled England for two years before being succeeded by Henry VII after his apparent death. (BFA: The Kingmaker, EDA: Sometime Never...)

16th century
Henry VII was the King of England until his death in 1509. (EDA: Sometime Never...)

Henry VIII was the King of England from 1509 to 1547. In his first incarnation, the Doctor claimed to have argued with Henry VIII so that he would be sentenced to the Tower of London where the TARDIS was located. (DW: The Sensorites)

After the death of Henry's successor, Edward VI, Jane Grey became queen at sixteen years as her father wished, although she never wanted to rule. On 19 July 1553, she met Rani Chandra, a traveler from 2010, on the ninth and final day of her reign. (SJA: Lost in Time)

Mary I had claimed the throne and proclaimed her a traitor, intending to sentence her to death. (SJA: Lost in Time)

Elizabeth I was the Queen of England in the year 1599, and an apparent spouse of the Tenth Doctor. After the Doctor broke off with her, she met a younger version of his tenth incarnation following the premiere of Love's Labour's Won, and was furious with him. (DW: The Shakespeare Code, The End of Time)

17th century
After the death of Elizabeth I some time before 1604, the Scottish King James I inherited the English throne. From then onwards, both kingdoms were ruled by a single monarch. (VG: The Gunpowder Plot)

19th century
Victoria, Queen of Great Britain and Ireland, Empress of India, usually known as Queen Victoria, was a powerful and influential monarch on Earth in the 19th century, ruling from 1837 to 1901. Her reign, known as the Victorian era, was marked by tremendous strides in technology and the arts in Britain and elsewhere. Victoria met the Doctor at least twice: once during her coronation in 1838, and in 1879 she shared an adventure with the Tenth Doctor. The latter encounter led her to banish the Doctor from her country and form the Torchwood Institute, as the British Empire's defence against the dark forces with which the Doctor seemed to consort. (DW: Tooth and Claw)

20th century
Edward VII was King of Great Britain from 1901 to 1910.

The Doctor claimed to have met Edward VII in Paris. (DW: Inferno)

Edward VIII was King of Great Britain for most of 1936. He chose to abdicate before marrying the twice-divorced American Wallis Simpson. (PDA: Players)

Elizabeth II of the United Kingdom, born on 21 April 1926, became the Queen of the United Kingdom in 1952, coronated in 1953, (DW: The Idiot's Lantern) and served well into the 21st century. (DW: Voyage of the Damned)

The alien Wire attempted to take the energy from the millions of humans watching her 1953 Coronation on their televisions. However, the Tenth Doctor foiled the Wire's plans. (DW: The Idiot's Lantern)

Students at Brendon Public School (as well as the rest of Great Britain), where the Brigadier taught, celebrated the silver anniversary of Elizabeth II's coronation in 1977. (DW: Mawdryn Undead)

In 1988, her path almost crossed that of the Seventh Doctor (who almost but not quite recognised her) at Windsor Castle as she walked her pet corgis about the grounds. (DW: Silver Nemesis)

Elizabeth II was concurrently the Queen of Australia, a situation with which not all Australians were satisfied. The Fourth and Fifth Doctor's companion Tegan Jovanka, who described herself as "downright Bolshie," favoured her country's transition from a constitutional monarchy to a republic. (BFA: The Children of Seth

21st century
In 2006, Harriet Jones was invited to form a government by the Queen. (WEB: UNIT website (Secure section)) On Christmas Day of that year, the blood control powers of the Sycorax (operating out of a ship hovering over London) threatened to make Elizabeth II, as well as the rest of the British royal family jump off a roof and kill herself. (DW: The Christmas Invasion)

After the Battle of Canary Wharf (DW: Doomsday) in 2007, the Queen ordered the immediate closure of Torchwood One. (WEB: torchwood.org.uk) The level of her involvement with and support of the Torchwood Institute, established by Victoria, both before and after the Battle of Canary Wharf, is unclear.

Due to the Sycorax and "Christmas Star" incident (DW: The Runaway Bride) occuring on consecutive Christmases in London, most of the city's residents fled during Christmas 2008. The Queen staunchly decided to remain in Buckingham Palace. On Christmas Day, the alien Titanic almost crashed into Buckingham Palace; the Tenth Doctor telephoned the Palace and spoke a pre-arranged code number that resulted in the immediate evacuation of the Queen and her staff, but at the last moment he was able to pull the ship up above the palace. The Queen thanked the Doctor by name. (DW: Voyage of the Damned)

In Easter 2009, the Tenth Doctor left his TARDIS in Buckingham Palace gardens, telling Captain Erisa Magambo that the Queen did not mind. (DW: Planet of the Dead)

Britain had a king by 2012. (EDA: The Shadows of Avalon) Queen Elizabeth was eventually succeeded by Charles III, along with his Queen, Camilla. Charles himself was succeeded by William V. (QR: Revenge of the Judoon)

In 2050 when the British government was controlled by Lomax, a Korven from the Far future, Britain (and possibly other British government territories) was ruled by an unnamed King. An Oroborus arrived on Earth around his birthday, causing time distortions which meant that some people complained about missing the King's birthday message, and others saw it twice. (K9TV: Oroborus)

Far future
Elizabeth X of the United Kingdom, (also known as Liz Ten) was the Queen Regnant of the United Kingdom of Great Britain and Northern Ireland during the time during which Earth was devastated by solar flares in the late 30th century. Having made herself immortal (or near-immortal), she lived for centuries (DW: The Beast Below) and lived until at least the 52nd century. (DW: The Pandorica Opens)

At an indeterminate point in the far future, the British monarchy was deposed and the last monarch was executed. Footage of that event could be viewed through the Gogglebox located on the Moon. (BFA: The Reaping)

Alternative 20th century timelines
In an alternate timeline in which Germany won World War II, Edward VIII was restored to the throne with Wallis as his Queen in 1940. He signed a treaty which established Great Britain as a protectorate of the German Reich. (NA: Timewyrm: Exodus)

The regime of a fascist Britain had the royal family executed prior to 1943. (DW: Inferno)

Edward IX was King of England in a more technologically advanced parallel universe. (TVA: Who's Who?)

Alternative 21st century timelines
By 2007, Elizabeth II was not on the throne in Pete's World; on that world, Great Britain was a republic governed by a president. (DW: Rise of the Cybermen)

In a nightmare alternate timeline created by the Trickster, the starship Titanic did hit Buckingham Palace and caused a nuclear explosion. The Queen was presumed dead. (DW: Turn Left)

Alternative futures
In 5145 in the Time Field timeline, Elizabeth X caught River Song attempting to steal Vincent van Gogh' painting of the TARDIS exploding. After holding River at gunpoint she saw the painting and learned whom it was meant for before letting her go. (DW: The Pandorica Opens)

Other universes
In Arthur's World, a parallel universe, Arthur was King of England in the 8th century. (DW: Battlefield)