Are there Japanese descended people in West Coast of Britainia? How are they viewed by the general populace and how would they be viewed after a hypothetical war with Japan? What happens to soldiers of various ethnic wars when Britainia goes to war with their ancestral homelands? Are they reassigned to different theaters of war or are they absorbed into intelligence services? Also what would happen in a situation like the real life Akune brothers where two fought for the US and 2 for Imperial Japan during the war? I assume in a Britainia version of events two brothers would be shot for treason but how would the rest of the family be viewed by Britainia?
So, talking about race and ethnicity in the armed forces of the empire...
Well, first, let's get royal battalions and noble guards units out of the way. They're highly irregular and weird because they're run almost completely on the prerogative of the noble or royal paying for the unit(s). That means each formation works under nearly-unique rules, though most follow traditions based on the actual imperial army/navy/marines/air force, largely depending on the whim of the blueblood in charge. Although this is usually considered armed service, units like these are only formally incorporated when a noble or royal has official standing within the main imperial forces themselves.
It's
entirely up to the head of the unit as to what rules they set about race, religion, ethnicity, ancestry, etc... Hell, there's nothing stopping a noble from requiring all recruits in their guard units to be left-handed if they really want to. This usually isn't a problem given that guard units are almost always domestic, meaning that they're barely ever deployed in foreign conflicts.
This is somewhat less true for Royal units such as yours, or Cornelia's for that matter. Again, it's entirely up to the noble as to whom they employ as soldiers, but Britannia has a long and storied history of picking out oppressed minorities in the areas they invade and offering them guns and training to turn against the people in power. They know the land, they know the language, they know all the weak points in the government's structure to hit.
What you've done with the shinobi of Japan is only remarkable as a matter scale. No one has likely ever recruited such a large, well-armed, and already militant-minded population, but the theory is the same, just...
bigger. Usually, though, units composed of foreign troops, when they're fighting against their former friends and neighbors are a little more delicately managed than your shinobi. They usually take up the non-critical support positions that let more tested fighting units focus on the actual fights. They're also given overseeing officers and a command staff that are multi-generational Britannians who have a proven history of loyalty to the empire. Much like what's talked about in the video you've linked to above. How much discrimination there is varies wildly from unit to unit. Usually, though, it can be assumed that new recruits from 'militia' battalions raised from locals that you're invading are going to be given the most dangerous work and the most tedious, back-breaking assignments to 'prove their loyalty' as something other than would-be spies.
All of this applies to units in the regular armed forces as well.
The Imperial Army, Navy, Air Force, and Marines don't tend to raise native units until the region has been sufficiently pacified that a Viceroy can be installed. At that point, it's
their responsibility to do so. The imperial armed forces try very hard to only take soldiers that have had full training and screening. Britannian units from the formal army aren't usually organized along racial or ethnic lines, but neither are they explicitly mixed.
What this means is that a given unit's racial, ethnic, and/or religious composition really depends on where its soldiers were recruited from and trained at. So, taking the Philippines as an example, a unit raised there will likely be composed almost entirely of ethnically - Filipino individuals whose primary religion is Catholicism (heritage of Spanish colonization), and who likely speak either Spanish or Britannian English as their main language, with the other as a secondary tongue. The officer corps would almost certainly be composed of lesser nobility who'd been given tracts of land on the islands as rewards for military service. These would likely be either ethnic Britannians (white anglosaxon protestants) or perhaps the children of talented officers recruited from the armies raised from South America and South Africa used to invade the Philippines twenty-thirty years ago. There might also be a visible, if not significant, minority of wealthy Filipinos whose families had managed to maintain their wealth during and after the empire invaded. Or, rarely, been able to profit off the empire to the point where they could spare the funds to buy a child a commission in the armed forces.
Now, all of the abovementioned troops are going to, effectively, be second-generation imperials. Their parents may have lived through the conquest of the islands by Britannia, but they've grown up in the shadow of the empire. These troops are the ones that served as the vanguard of the conquest of North Africa.
This is the crux of Britannia's long-term strategy to keep control of the empire. It's to use armies raised from their previous conquests to conquer the next territory and continue the cycle. This displaces a generation of the youth, indoctrinates them to Britannian military discipline, and rewards the most talented of them with land and wealth from a distant region, encouraging them to relocate to the newly-conquered areas where they'll be more dependent on the empire's enforcement of law and order to keep what they've earned. The remainder who don't earn land and titles will return to their home territories, but now be considered imperial citizens instead of merely honorary citizens, receive greater benefits, and be on call in the event of an uprising against the empire. Everyone wins.
Except for people who can't or won't joint the military and are relegated to second-class citizens to suffer poverty or the indignity of being employed as cheap menial labor.
Those people lose, hard.
The case of the Akune brothers, in particular, would not be what you think at all. Britannia would require oaths of loyalty and they would be watched by intelligence operatives, but there's every chance that such a significant incident of loyalty to the empire rather than their families or ancestral home would be greatly rewarded. They would probably be
treated like show dogs, trotted out for display and propaganda purposes, but they'd receive quite a bit of special treatment.
Now, all of this isn't to say that the empire has magically done away with the glass ceiling. The upper limits to advancement exist in a quite substantial and significant way. It's just that they're not so much ethnic or racial or religious. Don't get me wrong, being of a like mind to the ruling class
helps a great deal, but by and large the deciding factors are blood lineage, position of birth, and actual competence. Not necessarily in that order all the time, but they're the winning ingredients to advancement in Britannia. If you're lacking one or more, what you have on offer needs to be of even higher quality.