Tempting, but highly unlikely. There certainly wasn’t any India Indian delegates addressing parliament, that I am aware of, and the situation would seem to be at least in part analagous… strong tribes/nations would have been given some autonomy, in exchange for trade rights and assistance with becoming “civilized”.
I do like the idea of the Queens 1st Apache Light Infantry though. 
There were some Indian-born Indians elected to the House of Commons as members representing English constituencies, such as Mancherjee Bhownagree and Dadabhai Naoroji.
It’s fascinating to me that they were both Parsis, considering how few Parsis there are.
Aren’t Parsis stereotypically known as being businessmen, lawyers, and academics? These are the kinds of professions that would create political leaders.
Parsis were prominent in trade and industry at the time. Even today, Bombay is the center of Parsi society in India. It’s not all that surprising to me that they would have disproportionately close relationships with the English at certain points in history.
That and rockers.
Britain had an empire in Africa and Asia or is India and pakistan not in Asia. Thay had a colony in Soth africa and Egypt or are they not in Africa.
I suspect you didn’t quite understand the question in the OP. What would have happened if the British had not lost the American colonies?
What happens to the southern U.S. (or whatever one calls it in the alternate history) when the Slavery Abolition Act of 1833 gets passed? Do they get an exception, like the East India Company territories?
In The Two Georges, slavery was abolished in America along with the rest of the Empire in the 1830’s. There was a lot of protesting in the American south but Andrew Jackson was the Governor General of North America and he squashed any attempt at rebellion before it could become serious.
In The Two Georges, the American colonists negotiated some unspecified settlement for their grievances that did not include representation in Parliament. If it had included that, abolition of slavery by Act of Parliament might have . . . taken longer. The more so as such a deal would also have necessitated giving Parliamentary seats to the slave-sugar islands of Jamaica and Barbados.
Think about that. America and Britain would have hashed out the slavery issue together. American Abolitionists and British Abolitionists would have worked together, on the ground and in Parliament; by the same token, slaveholders would have found their allies in Britain (beginning with parties financially interested in the trade). I.e., it would not have been an America vs. Britain thing. No matter how it came out, the process would have bound the Empire closer together, in terms of political culture. The idea that some issues cut across regional interests would have been planted early.
And imagine if Kipling had grown up the son of an Imperial official in the Wild West . . .
The British may have wound up weaker for it. One thing the American Revolution showed the British was that they didn’t need to control colonies to profit from them. They still got rich trading with an independent America and didn’t have to pay upkeep.
This was a strong argument for the Spanish colonies in South America keeping their independence, British trade. The UK navy wouldn’t let country stop their trade. They didn’t care who ruled it, so long as they could profit from them.
The British colonial strategy went from colonies to key colonies to defending their trade routes.
In a practical sense, the slave-owning planters of Jamaica and Barbados did have seats in Parliament. The British Parliament of the 18th century was pretty wide open to “special interests”. The Caribbean plantation owners sent money to parliamentary districts in England so they could choose the MP’s for those districts. Those MP’s then represented the sugar interests in Parliament. (And one example of this was during the Revolution. When France declared war against England in support of the Americans, the sugar interests arranged to have a large portion of the troops and ships in America pulled out of the United States in order to defend their islands from French raiders.)
During the crisis leading up to the American Revolution, a lot of political insiders in London took Americans in the city aside and pointed out this was how things worked. The Americans should stop complaining about not being represented in Parliament and start buying into the system. It could have happened.
Another possibility would have been the British giving the Americans a fair and open deal - let them have direct Parliamentary representation. Then let the numbers sink in - there were less than three million Americans in 1776 (including slaves) and over six million Britons. The American could have had the representation they had been asking for and then gotten the taxation anyway. And they couldn’t have even claimed it was unfair - taxes in the UK were already higher than the taxes the Americans were protesting about.
At first. But, eventually, at some point in the 19th Century, differential population growth would transform the British Empire into an American Empire. 
There might even be a bill to move Court, Government and Parliament to New York . . .
I don’t know. I think tradition would keep it in London. Nobody’s suggested moving the American capital to California.
I recall reading there was a bill in Congress in 1865 to move the capital to St. Louis, as a more central location.
Of course, if the British Empire in our timeline had been organized that way, it would have been effectively an Indian Empire.
That’s another thing. Whatever settlement the Crown reached with the colonies, it would set a precedent for future colonial acquisitions. E.g., if the Americans got seats in Parliament, it would not necessarily require every colony acquired thereafter got seats – but the distinction between those that do and those that don’t would stand out glaringly, and there would be a lot of political pressure to enfranchise all of them. Followed, eventually, by the pressure to extend the vote to nonwhites, and then things really get interesting.
The British Empire historically made that distinction between the “white dominions” that had a large degree of autonomy and the non-white colonies which were under more direct control.