Well taking a broad speculation, where Ernest Augustus, maybe with the Duke of Wellington as PM, basically try to maintain a High Tory rump and are forced to flee during an uprising, you'd probably get a mishmash of results.
I think you'll need a middle-class and working-class alliance for a relatively simple revolution, it will also see lofty ideas beyond cheaper bread and full employment come in. remember the Chartists were more an 1840s thing, brought about by the middle-classes being placated by the 1832 Reform Act. Avoid that and they remain united.
I think radical overhaul of constituencies, universal manhood suffrage (or rather household suffrage), abolition of the Lords, replacing it with a Senate I imagine, though they might go unicameral. Removal of tariffs, depending on the make-up of the revolution, independence for Ireland (though possibly as a de facto satellite of Britain) or if not, the new Republic has its first war on its hands.
I actually doubt the HEIC would 'secede' from Britain - there's still plenty of middle-class self-made man types in the Company hierarchy and I imagine they'll want British markets over loyalty to some reactionary German nut job. I can see some of the princely states using the situation to revolt but the situation on the ground hasn't changed so I think at worst it will be a smaller *Indian Mutiny.
Given the Westminster system that will probably maintained in some form, I can see a ceremonial President with long terms being put in place as a unifying figure head, to oversee a Parliament that might very well have annual elections though I doubt this, they could still be 3 or 4 years fixed terms.
Canada is going through its own revolts at this time - however I can imagine it will be complicated. whats to say a strong Royalist faction don't declare for the King? Republic tells them to go stuff it, local nationalists rise up and then the Americans see whats going on and might take advantage. Maybe they seize them or set up puppets, or maybe the Republic sorts it out and establishes a dominion thirty years early? Of course without a monarch such a state might be either a more equal free state in alliance or an autonomous part of the Republic.
Apart from that I can't see the colonies being let go.
I imagine for revolution to hit Britain, the King will have to had annoyed a lot of people and if its a 'American' style affair, I can see large parts of the Army and Navy joining in or simply sitting by while the King is booted out.
I can imagine a lot of people expecting a reformed monarchy however, with Ernst Augustus in good health and with a power base in Hanover, I can see the Republic slowly forming for practical and ideological reasons. EA's son George wasn't much of a liberal himself so you've got a couple of generations for the Republic to become entrenched.
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ANYWAY - the OP: I think republican brotherhood will strengthen the good old humour if there's no Canadian War, at least at first. However there's a good chance populism will have a greater impact on British politics in such a world. Say the ACW breaks out. Say a *Trent Affair happens. Well the likes of Palmerston might just get their wish and declare war on the Union.