People who are wondering what the British would do about the Iroquois need not speculate, The Iroquois, although mostly neutral, did side with the British somewhat in the French and Indian War, and were rewarded for their loyalty with exactly the same treatment that they would receive from the continental authorities years later. Keep in mind British North America (i.e. Canada) stretches from "sea to shining sea" just as much as its non-commonwealth neighbor to the south.
We assume the Brits were more friendly to the Native Americans primarily because of British edicts that guaranteed lands west of the Appalachian Mountains to the Indians, and forbade white settlement in those areas. But this was designed to help the colonists, not the natives, because the British government could not afford to protect colonial settlements west of the mountains. British settlement in the less successfully uncontested Australian outback and Canadian prairie was not similarly impeded. The British as recently as the mid twentieth century encouraged the white settlement of Kenya, South Africa, Rhodesia, and various other places in Africa.
So don't assume that an America ruled from London would be necessarily more amenable to a Native republic.