Either the focus of the country would shift towards North America or North America would become independent. The majority of the white population of the empire would be living in the Dominion of America. In OTL, the USA surpassed England in the 1810s, and would either need representation in Parliament, eventually leading to dominance, or enough autonomy to basically be independent.
The American dominions would of course become more and more influential in time, but there's still the rest of the empire in existence, particularly the Caribbean and India at this time, plus any further gains in Asia, Africa or, likely, Latin America. The sectarian differences between north and south also have no reason to suggest that the American vs European split would be the dominant one that mattered.
In the modern USA, would we say there is a strong feeling that the Eastern USA dominates the West? If you calculate it arithmetically, it is definitely true, but nobody particularly notices it. In this trans-Atlantic 19th C British Empire, it's possible the rural populations West of the Appalachians complain about those metropolitan businessmen in New York and London in the same breath. Meanwhile abolitionists in London and New York seek common cause against the slave power in Kingston, Havana and Atlanta.