It may not have been anything major but Anglicans did split from the Church of England after the war and formed their own churches.
And I guess it may have influenced some more.. hardline revolutionaires before/after the war to just change of churches maybe for things as Methodism (if it existed already?) because it was THE official british church and all. Or maybe I speak through an hat, to use a french language expression...
Hey, I was told americans switched tastes to coffee from tea in parts due to the war, according to pop history, after all... so, churches maybe... seen as a 'Whore of London'...
Maybe without ARW, Anglicanism would remains a bigger part of the Protestant religious side of things, and somewhat 'official' if less to the degree of now modern Canada.