Quite hard, considering how IOTL the Japanese tried so hard at the Koreans and the British tried so hard at the Irish. A primary necessity is the fact that these "new" cultures can be introduced as equals into British society/culture.
Seems to me that this example of Ireland is very apt. Why would English culture blend with Indian and/or African cultures before Irish, who were much closer linguistically, culturally, ethnicly, religously, and poliically?
In modern times middle class Indians have some really rather British habits. Its not too ASB for this to become stronger and for the Indian influence on the UK to similarly be stronger.
Britain didn't really try with Ireland actually, it was the Irish themselves who saw more opportunity in adopting English who did their own integration.
In modern times a northern English/Scottish person has more in common culturally with an Irishman than they do with a southerner.
Politics and culture rarely follow the same borders.
Tyr has it to be honest - though it was worsened by a genuine English disgust/distaste for the Irish. If the English had been pro-Irish rather than just wanting to displace them, then I imagine that assimilation would have been quite comfortable.
If the British started respecting their colonial subjects, this could take place, but it would require some ASBs.
Great idea. A fusion of Indian and British cultures would be awesome. And what if the British allied with the Zulus in the mid-1800s, carving out a joint empire in Southern Africa. Perhaps an Anglo-Zulu dynastic union?
Having Zulu troops on the imperial side would be a great advantage, considering how tough and disciplined they were. Maybe they could convince the British Army to give their men uniforms that wouldn't make 'em die of heatstroke.
I've had this idea in my head of a steampunk Zulu Empire that imported engineers and teachers from Germany to build infrastructure. Imagine if the Zulus had gotten access to full European-style ordnance, Gattling guns, mortars, and everything.
Steampunk Zulus sound awesome, but I can't see it working - Shaka Zulu changed how Zulu warfare worked to be conquest, but utterly changing their style of war could be somewhat uneventful, as they don't have the infrastructure to support that style of warfare.
Though regarding respect, I think this is going to be the overriding issue. As much as this could butterfly the remaining Empire - if for some reason or another the larger Irish communities in America could be staunch loyalists that counter the revolutionaries? This would certainly endear them to the British proper, perhaps changing the Irish from "Damn Paddies" to "Good Irish Patriots" in the popular zeitgeist. If Irish is seen as similar but not the same, but capable of loyalty, then it could certainly encourage that attitude once the British go east.
Though an East India company that instead of getting repeatedly bailed out, is instead replaced with British vassals? If they intermarry (which isn't a bad idea) then in India an Anglo-Indian cultural fusion could begin in the upper class, influencing back home.
Put off trade companies holding territory and armies vs vassals, the British could then in Africa institute a similar policy, creating Anglo-African nobility. With the Anglo-African, Anglo-Indian, and Anglo-Saxon cultures all having to deal with each other in formal and functional situations - a sense of tolerance would emerge, and likely be fostered by the crown to keep them stable.
Admittedly, as Socrates points out below, marrying African tribal Kings is going to be very unlikely, but if they set up an African, educated and loyal, and Anglo-African, then they would be marriagable, but probably not by the monarchy, not for a long time.
The British monarchy marrying African tribal Kings, or Anglicanism merging with paganism centuries after its theology having been settled are alien space bats. I don't think people realise how entrenched the concepts of racism and British superiority. They were what justified empire in the first place and you'd have to go back to before the empire ever existed to butterfly them.
Hence -
Isn't Afro-Caribbean culture in a lot of the West Indies similar to what your end result would be? You could easily use that as an analog, seeing as I would expect some kind of creole language and mixed culture just like what developed otl in the Carribean.
Not in the least familiar with Afro-Caribbean cultures to be honest. Though I think your points on creole language and mixing would be true. Just take a look at Singlish in Singapore, they always form.
