In short, we're looking at a Britscrew scenario.
Not necessarily. We in the Foreign and Dominion Office have frequently wargamed this: it's almost as popular as Operation Lion d'Marine (What if Napoleon did invade?!) or Imperial War Z (Redshirt Zombies!). Consider the following...
The reintegration of British North America back into the Empire lulled the then-Colonial Office into a false sense of security: the secession of the Thirteen Colonies could be falsely dismissed as an aberration in the smooth upward flow of proper colonial administration. Consequently, the violent insurrections and sedition against HMG in Ireland, the Raj areas and the Upper Canada region in the middle of the 19th century came as a surprise when it shouldn't have, and the loss of the second Boer War just emphasised the point. Consequently the Dominion concept (extensive devolution of powers but retaining the Crown as head of state and Westminster as the supreme court) wasn't rolled out until the 1890's: as PistolSO points out, BNA didn't attain full Dominion status until 1897 when the American Assembly became the BNA Parliament and Teddy Roosevelt became First Minister. Would Dominions have arisen earlier if BNA had stayed outside the Realm?[1]
As SMJB correctly point out, the Raj areas didn't become Dominions until decades later and arguably the most succesful application of the concept is the Dominion of Australia, which was unified in the 1900's. It is fascinating to consider what would have happened if the economies of scale shown by a unified Australia were applied to a single Raj Dominion (Dominion of India?).
A more subtle point is the freezing of political development. The devolution of power to very-large subunits whilst retaining a central head-of-state has proven stable, peaceful, and democratic, and it has been copied world-wide: witness the Kaiserein's recent attempts to enable her Europäischen Gemeinschaft to meet its 100-yr anniversary by devolving power. And we in the Foreign and Dominion Office are content with this.
But we cast our mind back to 19th century scientific romanticism and their postulated systems of government. Would the Parisian Communards have spread their "Universal Socialism" as wide as they dreamt? Would Jefferson's dreams of a Roman Republic on the Potomac have lasted? We abolished slavery[2] and created law and moderate government under God and the Queen for all male landowners over the age of twenty-five. Was that the best the universe could provide?
[1] OOC. IOTL the Dominions started being created in the 1850s to prevent the successful secession of another colony after the Thirteen demonstrated what can happen if you don't pay attention to the governed. If you delay or abandon the devolution inherent in the Dominion concept, the Empire eventually cracks. Note that ITTL the Boer Wars were lost, so the Dominion of South Africa is smaller and never develops apartheid
[2] OOC. ITTL overt racism is abhorrent and illegal in this Dominion Civil Service: all may enter and take the Imperial Examinations and succeed by their own wits. Covert racism, caste systems, voluntary separation on the other hand...