The Answer: Maybe, maybe not.

The name Takako seeems to be a modern name. Women in Middle Ages Japan mostly had 2 (or 1) Japanese syllables in their names. For example, see the register of names from the official investigation conducted in 1671 below:

A sampling of the first 5 names from the top row: A-Ki, Ka-Me, Fū, Man, and Ka-Me. Out of 127 women in a Middle Ages village, only 10 women have 3 syllables in their name.

Ta-Ka-Ko sounds to me like a modern name after Japan "opened" and revolutionized in 1868, when Japan officially allowed peasants to have names. (Prior to that, some Japanese peasants did not even legally have a "name".)

But it is true many Japanese peasants were "traded" to Europe by the merchants who came to Japan with Jesuits in 16th century.

And regarding what you mention in the quest:

The apparent timespan puts Takoko's presence directly at the beginning of Japan's Sakoku, their period of severe isolationism, and in fact Britain had cut off all trade with Japan in 1623.

I don't know where you got that information, but on 8th January 1614, the Tokuguwa shogunate officially announced the complete ban on the spread of Christianity anywhere in Japan. It is called 伴天連追放之文 (Bateren-no-Tsuihō-no-Bun), per this Japanese Wikipedia article; the corresponding section of this English Wikipedia article translates it as the "statement on the 'Expulsion of all missionaries from Japan'".

P.S.: The "official" but last "English" advisor, Willam Adams from Britain, had already died in 1620.

P.S. 2: Already before the Tokugawa family seized power, 26 Japanese were crucified by Hideyoshi Toyotomi on February 5, 1597; they became known as the 26 Martyrs of Japan.

Question:
Was it possible for a young Japanese woman to end up enslaved in Great Britain in the mid-1600s?

Possible? I would say yes. While it's true Japan's official slave system from the Yamato period (3rd century A.D.) through the Toyotomi Hideyoshi was abolished in 1590; the Western definition for slavery is perhaps broader in nature than that which is employed and banned in Japan in the late 16th century.

see Japan's encounter with Europe, 1573 – 1853

Forced or "Non - Free labor", described as Gotōke reijō (Tokugawa House Laws) article 17, identifies a form of slavery of the immediate family of executed felons. This practice persisted in Japan from 1597 and 1696. While uncommon there are over 600 documented occurrences of this practice over 101 years of the seventeenth century.

Likewise while it's true, all Portuguese were forbidden from entering Japan in 1639, after a failed Christian uprising. The Portuguese were not the only Europeans in Japan. The Dutch had arrived in Japan in 1600, and remained in Japan after the Portuguese ban had taken effect.

see Slavery in Japan

It is not impossible for dutch trade to have included a Japanese slave in the 17th century. The Dutch were active in the African slave trade from 1612 and 1872, and did provide slaves to British colonies. While Britain never technically legalized slavery domestically, worst case one could say slavery was widely tolerated in the domestic UK in the 17th century while of coarse being mandated in it's colonies.

Admiral Sir John Hawkins of Plymouth is widely acknowledged to be "the Pioneer of the English Slave Trade" (from 1554–1555). In the 18th century slavery became a major component of the British economy, and in the mid 1700's Owners of African slaves in England would advertise slave-sales and rewards for the recapture of runaways.

Cromwell in the mid 1600's sold large swaths of the UK Gypsy population into slavery, as many as 50,000 Irish were sold into slavery after the Irish rebellion of 1641. It's "possible" a Japanese slave transported by the Dutch could have found his or her way to the UK in the mid 1600's.

see Slavery in Britain

Sources:

• Japan's encounter with Europe, 1573 – 1853
• Slavery in Japan
• Slavery in Britain

Short Answer:

That story is not impossible in the sense of violating laws of physics, but extremely improbable, especially as some of the characters may have violated the laws of one or more nations and risked severe punishment by doing so.

Long Answer:

in the 17th century (1601-1700) a number of English persons were slave owners, of a sort. They were investors in various large or small companies that participated in the Atlantic slave trade. Those companies sent ships to Africa to buy African slaves and ship them to North or South America to sell as slaves.

Technically those companies owned the slaves that they bought and sold. Individual investors would not own the property of the company as individuals, they would have a share in the company which owned property. If a company was owned by one person that one person would own all the property that the company owned, including slaves.

But slaves were purchased in Africa for the purpose of being sold in various colonies in the Americas, and not for being used in Britain or England.

Of course, there were a number of English colonies in the Americas. Slavery was legal in those colonies during that period and until 1833. And some of those English colonies, full of English colonists, the ones in what is now the southern USA, and in the Caribbean, had plantations with large slave work forces.

Most of those plantations were owned by English colonists. An English colonist who became wealthy and successful in the colonies, or their child, might move back to England and enjoy a wealthy lifestyle there. And they might take their personal house servants back to England with them, including slaves. But anyone who took a slave to England would run the risk of having that slave declared free, so I don't know if that was ever done.

I do know that some English aristocrats in that era had black servants, and I don't know if they were legally slaves or legally free.

During and following the Renaissance, it became fashionable for black boys and young men to be decorative pages, placed into fancy costumes and attending fashionable ladies and lords. This custom lasted for several centuries and the "African page" became a staple accoutrement of baroque and rococo style.[5]

https://en.wikipedia.org/wiki/Page_(servant)1

If the fictional character Takako was old enough to remember cherry blossoms and chrysanthemums when she left Japan, she would probably be at least three. If she was still young when killed, she would have been under forty, so she should have left Japan no more than 37 years before being killed.

The latest time that Takako left Japan legally would be before it was forbidden. The Japanese policy of isolation was establish by several edicts between 1633 and 1639. The edict in 1636 established the death penalty for Japanese leaving Japan. So if Takako left Japan before 1633 or 1636 at the very latest, she should have been killed by 1670 or 1673 at the very latest.

So it is possible that Takako left Japan by 1633 or 1636 at the very latest, if she left legally and not as a criminal, perhaps fleeing from the law, and possible that she left Japan as a young child, even though it would be unusual for children to sail overseas. And possibly Takako was enslaved somewhere, and sold to Mr. Ermengarde somewhere in Asia, and Takako might have been a companion to Charlotte Ermengarde if they were children of a similar age.

One romantic possibility for Takako fleeing Japan after 1636 would be if she was a relative of Amakuso Shiro or someone involved in the Shimabara Rebellion of 1637-38.

And the Ermengarde's might have returned to England after successful trading in Asia, perhaps in British trading posts in India. And after her parents died, Charlotte Ermengarde might have declared Takako legally free, since after all Takako's status as a slave would be legally dubious in England. If Takako worked for Charlotte as a servant after being set free, Charlotte would have had a large amount of power over Takako according to the customs of the day.

Of course no mistress of a household had the legal right to order someone, even a servant, executed. Only a magistrate could order someone executed after they were convicted of a capital crime in the magistrate's court. And presumably there were no female magistrates in 17th Century England.

Edward Wightman, a Baptist from Burton on Trent, was the last person burned at the stake for heresy in England in Lichfield, Staffordshire on 11 April 1612.[78] Although cases can be found of burning heretics in the 16th and 17th centuries in England, that penalty for heretics was historically relatively new. It did not exist in 14th-century England, and when the bishops in England petitioned King Richard II to institute death by burning for heretics in 1397, he flatly refused, and no one was burnt for heresy during his reign.[79] Just one year after his death, however, in 1401, William Sawtrey was burnt alive for heresy.[80] Death by burning for heresy was formally abolished by King Charles II in 1676.[81]

The traditional punishment for women found guilty of treason was to be burned at the stake, where they did not need to be publicly displayed naked, whereas men were hanged, drawn and quartered.

There were two types of treason: high treason, for crimes against the sovereign; and petty treason, for the murder of one's lawful superior, including that of a husband by his wife. Commenting on the 18th-century execution practice, Frank McLynn says that most convicts condemned to burning were not burnt alive, and that the executioners made sure the women were dead before consigning them to the flames.[83]

The last person to have been condemned to death for "petty treason" was Mary Bailey, whose body was burned in 1784. The last woman to be convicted for "high treason", and have her body burnt, in this case for the crime of coin forgery, was Catherine Murphy in 1789.[84] The last case where a woman was actually burnt alive in England is that of Catherine Hayes in 1726, for the murder of her husband.

https://en.wikipedia.org/wiki/Death_by_burning#England2

So if Takako committed heresy (before 1676) or petty treason against her superior, she could have been legally sentenced to be burned at the stake.

As someone from Japan, Takako might have been raised as a non Christian (though being a Christian would have given her a strong motive to flee from Japan during that period when Christianity was outlawed) and her Christian beliefs might thus have been rather heretical, or might have been a Roman Catholic and thus a heretic in English law.

Once a girl from Japan gave a talk to my high school class. She said that her family were Christians. I wish that I had asked her whether they converted after Japan was opened up to the world, or if if they had been secret Christians for centuries when Christianity was persecuted in Japan. For all I know, she might have been related to Japanese Christian Martyrs. https://en.wikipedia.org/wiki/Martyrs_of_Japan3

As a servant, even a free servant, it would have been petty treason for Takako to kill her employer, and perhaps she killed one of Charlotte Ermengarde's parents, accidentally, in self defense, or murder..

But Charlotte Ermengarde would not have the legal right to sentence anyone to any form of execution.

So maybe Charlotte Ermengarde was torturing Takako with fire or red hot irons as punishment or to get her to confess to something, and Takako's clothing took fire and burned her to death before the fire could be put out. That would have been very extreme but probably not totally unheard of in 17th century England. Thus Takako might have been more or less accidentally burned to death.

Or maybe Takako was lynched at the orders of Charlotte Ermengarde, who thus would be getting away with murder for a longer or shorter time due to being rich and powerful. And perhaps Charlotte Ermengarde justified the lynching to herself and her accomplices by saying - accurately or not - that Takako was guilty of heresy or petty treason and would have been sentenced to burn by the courts anyway, so they were just saving the courts time and money.

In a comment Exal says that in the story Charlotte Ermengarde had the local court convict Takako of witchcraft. As wealthy estate owner she might have had a lot of influence on the local courts. Thus she would not be taking any risk by persuading the local court that Takako was a witch and getting them to execute Takako.

But the laws against witchcraft in England in the 1600s were the laws of 1563 and 1604, which made witchcraft a felony, which was thus tried by secular courts and not by ecclesiastical courts.

This provided, at least, that the accused persons theoretically enjoyed the benefits of ordinary criminal procedure. Burning at the stake was eliminated except in cases of witchcraft that were also petty treason; most convicted were hanged instead. Any witch who had committed a minor witchcraft offence (punishable by one year in prison) and was accused and found guilty a second time was sentenced to death.

https://en.wikipedia.org/wiki/Witchcraft_Acts#Witchcraft_Act_16044

So Takako would have been hanged if convicted of causing someone's death by witchcraft, or for a second offense.

But if Takako was convicted of killing someone in authority over her, she would be guilty of petty treason, and that would be legal grounds for burning her at the state.

Otherwise the magistrate who sentenced Takako would have greatly exceeded their legal authority by ordering death by fire and would face some possibility of being punished for it.

Thus it is seen that the account in the story is not impossible in the sense of violating any laws of physics, but is extremely improbable. Furthermore, some of the characters may have violated laws of Japan, or England, and/or other countries during the story, which is not impossible of course but does lessen its probability of happening.

1: Could there have been young Japanese women in Great Britain in the mid-1600s?

This seems extraordinarily unlikely.

According to the British Chamber of Commerce in Japan

1600 William Adams, a seaman from Kent, becomes the first Briton to arrive in Japan.

1832 Three sailors from Aichi Prefecture—Otokichi, Kyukichi and Iwakichi—cross the Pacific Ocean from Japan. After arriving in the United States, they join a trading ship that travels to the UK and, later, Macau. Believed to be the first Japanese to set foot on British soil, Otokichi becomes a British citizen and adopts the name John Matthew Ottoson. He later pays two visits to Japan as a Royal Navy interpreter.

(my emphasis)

A 1997 PhD thesis by Andrew Cobbing says:

Some communication with overseas traders was thus possible during the Edo period, but the sakoku. edicts also included a ban on overseas travel, and this severely curtailed opportunities for cultural contact with the outside world. The ban was imposed in 1635 and stipulated the death penalty for anyone leaving the country without permission from the bakufu. A number of Japanese returning from abroad at the time were summarily executed. Before the imposition of the sakoku edicts, the Japanese had been active in East Asian waters.

This suggests very limited opportunities, likely none, for a young Japanese female to leave Japan during the 1600s.

2: Would she have been a slave?

Very unlikely in England at that time.

Wikipedia seems to indicate that internal slavery in Britain was unheard of by 1200.

I see no reason to doubt this. The feudal roles of serfs, cottars and others probably provided the invading Norman lords with all the control they wanted over their subjugated workforce.

The Atlantic slave trade started in the 16th century, first by the Portugese and later by the British and others. But that was transporting slaves from Africa to America not to Britain and initially the victims of the trade had the rights of indentured servants.

So there certainly wasn't any well established system capable of enslaving citizens of Japan and transporting them from Japan, or elsewhere, to Britain.

Were there slaves (and markets where one could publicly free them) in Great Britain in the mid-17th century?

There were no slave markets where slaves were openly traded.

A few slave owning people moving to Britain from America could have brought some slaves with them and continued to treat them as slaves. This status would almost certainly not have been supported in English or Scottish law.

As we know today, you can still be a slave in a country where slavery is illegal. Just not openly.