I think we can play with the numbers here...
For starters, let's have Britian retain control of both North America and India. (Yes, I am ignoring the 1790 POD of the OP on purpose)
We could also have greater migrations into North America from Europe esp. Germany and Italy... just have them be disunited and war ravaged for longer, and let's say that NA also has the most progressive policy toward European Jews, and no discrimination against Catholic, Orthodox and Eastern European immigrants. And finally, no act excluding Chinese and Japanese from immigrating. Then Give them better fertility rates (probably with a form of protestanism with srong emphasis on procreation and "be fruitful and multiply" that trickles down to rest of society).
It will be massively helpful to avoid OTL wars (American Revolution, War of 1812, Civil War and the Indian Wars), with the corresponding results being better treatment of Natives and Freed Blacks, with slavery ending Earlier. No Trails of Tears, or Plague Blankets, no Wounded Knee, etc... so the native American population is higher too, and for the most part well-integrated through intermarriage to whites. No black emigration to Liberia, and better fertility rates among Blacks because of them being freed earlier.
On the Indian side of things, we can have the princely states excluded from the union (After all, they are merely protectorates) and have some annexed states on the fringes of the Raj (Sikh Empire, Sindh, what is now NWFP, Sikkim, Burma, Kashmir) left alone or as preotectorates-- essentially getting the same treatment as Nepal and Bhutan got OTL.
We can then, probably do a 50: 50 population balance of non-subcontinental and subcontinental peoples in this grand union of Great Britain, Ireland, North America and India. If we can have Indian middle classes sufficiently evangelized, modernized and English-speaking and a non-racist culture at large, then this might be a platable union for most people concerned.