I would agree that this is highly implausible. The French had a very good run in colonizing the Americas, along with the Russians, and the Dutch certainly were also definitely working. Beyond that, the Swedish and Danish also certainly had a shot, and theres no reason to presume that even a Portugal where the reconquista never happens wouldn't have an interest in the Americas.
Even if we somehow managed to get all of those out of the picture, we're still left with the problem of actually getting Britain to conquer all of the Americas. The Spanish had to go through a lot of implausible stuff just to get as far as they did, as did each of the other powers. The Natives aren't simply going to roll over in the face of British aggression. I highly doubt that Britain had the resources or will to conquer all of the Americas. Even with all of the 500 years you've given them, it would be a huge challenge. I just don't think this scenario is plausible.