If Mizu reaches England and kills the rest of the names on her list, what then? The isolation rules of the Sakoku period were reciprocal: Not only were outsiders restricted in how they could access Japan, the Japanese were restricted in how they could access the outside world. By leaving Japan without leave from the Shogun, Mizu has committed a crime punishable by banishment - if she ever returns, she can be executed on the spot.
Mizu runs on Revenge Before Reason, so likely she's either not thinking that far ahead or assuming she can solve that when the time comes. Certainly there are ways around that. She saved the Shogun's sons, they don't necessarily know she left the country, the government might not want to admit that the Sakoku policy wasn't as strict as they pretended, and if it comes down it, executing Mizu is easier said than done.
Does she care enough to return? Is there anything for her to really return for? She lives for revenge, and has no real ties to Japan. The only thing she may return to Japan for is to claim her reforged sword, which she'd only need if she has something else to live for. It wouldn't be that surprising if after killing all the "White Devils", she killed herself as well.
Mechanics of Slave-Trading
So, Fowler says that two of the other White Devils made their fortune by enslaving Japanese women and selling them to the English. But... how? Japan is on the far side of the world from England - with how transcontinental travel was in its infancy at this point in time (for reference, it took the expedition William Adams was a part of nearly two years to travel from Holland to Japan and practically all of them died in the attempt) it would be impossible to make any kind of slave-trading endeavour at all profitable this way, if it was even possible in the first place.
Given that Japan's history deviates from real life, it's possible that in this world, England had managed to advance its naval capacity to unprecedented levels in the 1600s with the possible cause that the kingdoms of Great Britain (England, Scotland, Wales) had just unified a century earlier into the Kingdom of Great Britain instead of in 1707 which would have allowed technological advancement considering that Fowler refers to the British in the modern sense.
I don't know if that really gels with how the setting is generally 'like real life but". Especially considering how Fowler refers to the events of the Nine Years War in terms that indicate we're supposed to take it the same way as it went historically. Furthermore, I don't think it'd ever be profitable to set up a slave trade like this, even if BES's English have developed the steamship off-screen. Slaves in England at this time would've been bought and sold for their value as labour in the New World, as part of the fledgling trans-atlantic slave trade. What market would exist for Asian slaves as novelties or luxuries just wouldn't be big enough to recoup the cost of dragging them halfway around the world.
There was actually a large scale slave trade in the late 16th century, where Japanese people were sold into slavery outside the country and some (especially women bound for the sex industry) even ended up in Europe, but the trade was between Portugal and Japan rather than England. The enslaved people were mostly sold in Portuguese-colonized regions of Asia like southern China and Goa, so this could have been the White Devils' main approach to turn a profit and Fowler was merely exagerating how many women were sold directly to the English.
A good point, but considering how BES has yet to even mention the Portuguese or Dutch so far (them being medieval Japan's primary trading partners in real life), in favour of making it seem like Fowler and his English compatriots are doing it all, I feel like this one is just down to Writers Cannot Do Math.
Taigen's scarf
Why does Mizu take Taigen's scarf after she knocked him out? She already had a scarf, so why take his? Later, Akemi recognises that Mizu has Taigen's scarf, leading her to believe Mizu killed him, but other than there seems to be no plot based reason Mizu needed the scarf?
It was probably petty revenge for Taigen bullying Mizu when they were both kids.
Social Mobility
Akemi's grandfather was a pig farmer, now she's marrying into the Shogunate. What the hell? That level of social mobility is insane. It'd be a major scandal for someone to do that to the (effectively) royal family today, let alone during a feudal period. What did Daichi do to become a lord? Save Japan from Godzilla or something? Is this an aluminum Christmas tree? Is it based off an actual historical figure who somehow did this? Because it seems really strange. In addition, if it was always Daichi's plan to marry I to the Shogunate to take control after the coup, why did he ever consider Taigen as a marriage candidate to begin with? Was he only brought in on the coup last minute?
The series takes place in what effectively is a Alternate History version of Japan (divergences from Real Life 17th century Japan include the Itoh Clan holding the shogunate instead of the Tokugawas, lack of firearms usage, and the isolationist policy working differently). It's not a stretch to assume that the Tokugawa Shogunate's rigid class/caste rules don't exist in this timeline. Anyhow, Akemi implies that Daichi became nobility the same way Taigen was planning to: He became a swordsman, received a position of aristocracy fit of his service, and grew his fief from there.
Such extremely liberal social mobility feels at odds with what we actually see of the commoners and their lot in life. Taigen is also pretty extreme, but I can buy a swordsman raising through the ranks and joining the nobility for the sake of a story, but marrying into the shogunate within the span of a single generation breaks my suspension of disbelief.
In real life, Toyotomi Hideyoshi went from a peasant to the de facto shogun of Japan over the course of a single lifetime in the 1500s. Since the show is set in the 1630s, Akemi's grandfather likely lived around the same time as Hideyoshi, and its possible he and/or Daichi climbed the social ladder under similar circumstances.
was a part of nearly two years to travel from Holland to Japan and practically all of them died in the attempt) it would be impossible to make any kind of slave-trading endeavour at all profitable this way, if it was even possible in the first place.