Why don't Americans have British accents?

Accents and dialects arise when people are separated from their original population - like the Americans were separated from England. Their version of English changed over time until it became different from British English. America also has many people from many different lands, so that there are lots of different words and pronunciations added to the language.
I don't see why they don't. The British DID colonize America, and Canada!
The Australians and New Zealanders got them. Why?

America had a huge influx of non-British people, especially after the war of Independence. And I'm sure that since we did REBEL from Britain, a British accent and British mannerisms in general would not have made you popular.
I suspect that Australians had a predominantly British immigrant population until late in the 20th century. They had restrictive immigration policies.

I'm going to keep the other answer so I can answer it too. Here's the thing about accents: I'm Portuguese so I can give you my word of knowledge. I learned English at school but I didn't learn the English accent, or any other accents, you have, for example the Texan accent, but the way you say it doesn't make a language. Now, to answer the answer, I hope you're not American, because if you are you need to go to school. There was an influx, like you said, of non-British, but not after any war. I'm not sure if by this you are talking about the independence from England, if you are you're right, but there was no war, or if you talking about the civil war which would be incorrect. Secondly, about Australia, until the 20th Century it was more of a prison. Some Americans were sent there for breaking the law. They were excommunicated, so it's accent over accent.