The really silly thing about speculating what would have happened had the British won the War for Independence (AKA US Revolutionary War) is that there has been a British North America: Canada. Books such as For Want of a Nail and the Two Georges seem to miss that point.
Did the Tories want to stay dependent to Britain, or were they interested in a peaceful transition to independence? The problem was that the “patriots” in the War for Independence were rather hot headed and intolerant of other opinions. There wasn’t a debate as to the prudence of armed rebellion. And it was hard to go back one blood was shed.
Canada did get its independence, but Canada’s transition from a self-governing British colony into a fully independent state was a non-violent, evolutionary process that came about in such a gradual fashion that it is impossible to say independence came on a particular date. When the Dominion of Canada was created in 1867 it was granted powers of self-government to deal with all internal matters, but Britain still retained overall legislative supremacy. Total Independence didn’t come until 1982 with the passage of the Canada Act by the British Parliament.
Like Britain, Canada’s Constitution is an amalgamation of codified acts and uncodified traditions and conventions which outlines how the government functions and the rights of Canadian Citizens.
The real question isn’t what would a British North America be like, but what would be its boundaries?