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?
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.