While waiting for a better answer, you may enjoy these.

Is the American Accent the real British Accent? - a take on the myth that the British accent is the one that changed (Spoiler - according to /u/TheAgileElephant, they've both changed from the common ancestor)

How did American Accents come in to existance - one from four days ago, asking about spelling and the accent and /u/Harsimaja discussing that

And then there's this one - How and when did the "American" accent come to be? - in which /u/lord_mayor_of_reddit gives an overview of earlier answers given.

Though I don't know if any of them will be capable of answering the question of "At what point did the distinct American twang appear?" as I don't know what you mean by "distinct American twang". There's at least 8 major dialects with their distinct speaking patterns and twangs, and most people associate Twang with one of the Southern dialects (and yes, there's multiple Southern dialects)