The headline question would be similar to “Did the Latin dialects throughout the Roman Empire evolve into different languages;” and to that question, I would answer most definitely they did and to the former question I would answer that they are already in the works of becoming different languages. In the case of Latin dialects, they would evolve to become French, Spanish, Portuguese, Romanian, and Italian. In the case of English, well, it might become British, American, and Australian (without the hyphenations).
There is a difference between a dialect and an accent. An accent has to do with the variation of word pronunciations that is unique from other accents of the same language; whereas a dialect not only involves pronunciations but also sentence structure. This article takes both of these into account, since they define the differences between the English spoken around the world.
As learned from my linguistics class, variants of the same language (in other words, dialects) can only become separate languages when there is geographical isolation from other variants and they develop independently. This would especially be true when if countries with English-speaking majorities are not just separated by land, but are separated by continents.
Oftentimes, the English spoken in any area of the world can incorporate words from other languages, with an example being South African English borrowing words from Afrikaans and other languages in that area.
There is already a moderate split between English in America and English in Britain, while their grammatical differences are subtle. Mixed with the Received Pronunciation taught by the upper class, British English has already become dramatically different from American English orthography, however intelligible the two dialects are.
If you spelled a sentence in its phonetic form, and it does not even have to use the International Phonetic Alphabet, in three English dialects, you would definitely conclude that you would need to use the English dialect you grew up speaking as a decoder. If that is the case for an English creole language like Gullah, then it would surely be the case for English dialects.
So far, only the English spoken in the English-speaking world have been mentioned, but not the English spoken all over the world. China, for example, is set to become the country with the largest number of English speakers. This mainly has to do with the appeal of the English language as a language of opportunity, specifically when it is the most widely spoken language in the world. To paraphrase Jay Walker, he explained that the appeal has to do with the fact that the English language is considered “the world’s common language for the world’s common problems.”