Several problems

"If an individual's children and father have an accent then the odds are very good that the individual himself will have an accent." Lots of improbable things happen in movies. The fact that it's unlikely isn't enough to make it a nitpick, though.

"Caractacus was born and raised in England so he definitly would have an accent." This has no basis in the movie. The movie doesn't say where he was born or raised.

One reasonable explanation is that Caractacus was raised for some of his early life in the U.S. or Canada. His mother (Grandpa's wife) might even have been American or Canadian. This would be completely within Grandpa Potts's character, since we know he is a great traveler. In the story portion of the movie, he sings a whole song ("Posh") about how much he loves traveling. At other times in the "real" portion of the movie, he's shown reading National Geographic, and talking about travels. In fact, just after Caractacus unveils the refurbished car, Grandpa emerges from the house wearing a fur coat and carrying ski poles, and says he's off to Alaska.

Grandpa not a reasonable source of info.

Grandpa is no where near a great traveler. During the 'real' portion of this movie he travels to Alaska, India, and Africa. All of which are located in his little hut. His "Posh" song tells that whenever he travels he travels first class (dining with the captain, etc...), just as he is at that point (standing in a roughly 3 foot by 3 foot hut, hanging under a Zeppelin, being dragged through the ocean, very posh). The only real traveling Grandpa ever tells about is with his brigadere (Lord Scrumptious).

\rThe hypothesis that Caractacus' mother was American or Canadian has no basis in the story and is therefore, I believe, irrelevant.

\rIt is pretty reasonable to assume that Caractacus lived in England his whole life. He is poor and very likely never had enough money to leave England.

Edition?

There seem to be at least two editions of this film. In one, there's more detail given, so it's clearer that Grandpa Potts is fantasizing about travel at present. However, it's still clear that he did travel in the past, as a batman in the army. And it's likely he traveled a lot, because he continues to think and talk about it. In any case, the relevant time is about thirty to forty years before, when Grandpa got married and Caractacus was born, and Grandpa would have been traveling internationally then.