Granted I'm sure the answer is probably going to be "A bit of both"
But where did that accent and it's derivations come from? Is it a mixture of remnants of the Brits and French that settled there with a dash of the US?
Before the Revolution, there were few English speakers who lived in mainland Canada. The population was virtually all French-speaking.
The maritime colonies/provinces were the only part of future Canada with a majority English-speaking population, and of those, Nova Scotia was the most populous. Nova Scotia had a population of roughly 12,000 around the time the Revolution broke out.
But even in 1767, about half that English speaking population of Nova Scotia was already American-born or American-descended. Little of the population was actually English-born. Nearly the entire population was either American, Irish, or German.
At the end of the war, somewhere north of 40,000 American refugees left the United States for free land in Canada, having been Loyalists during the war (1, 2). About half these refugees went to Nova Scotia. So by 1790, Nova Scotia had a rough population of 35,000, and about 25,000 of them were Americans.
New Brunswick and Prince Edward Island went through a similar situation, and since their populations had been much smaller than Nova Scotia's before the war, the Americans overwhelmed these colonies to an even greater degree.
Inland, about 10,000 Loyalists were resettled along the north shores of Lake Ontario and Lake Erie. This made up virtually the entirety of the population of Ontario ("Upper Canada") at the time. Before then, few Europeans lived in Ontario and the few who did spoke French.
The English-speaking population of Canada expanded from there. The country experienced a lot of European immigration during the 1800s and 1900s, but they were immigrating into a country whose English-speaking population was already established by ex-Americans, so that was kind of the "baseline" of what the English speakers would have sounded like at the time.
Of course, American accents, Canadian accents, and British accents have all changed quite a bit since the late 1700s. Maybe someone else can fill you in on how the accent evolved thereafter.
To answer your question, though, no the Canadian accent did not develop independently. It was more or less established by Americans who left the Thirteen Colonies. Further immigration from Europe and the United States, and regular interactions between Candian and U.S. populations affected it after that.
During the American Revolution, loyalists fled upwards to Canada, and began moving westwards. Through the generations, the children of the loyalists and other Canadians began to intermingle and adopt each others mannerisms, which led to the creation of the western Canadian accent.