The three original American dialects — New England, Mid-Atlantic and Southern — all had their origins in different areas of England and reproduce many of the characteristics of those areas. The varieties of Southern English, as well as the Midwestern twang, evolved in consequence of these dialects' rubbing together.
Non-English dialects have had relatively little influence, except in semantics and several structures which are clearly marked as non-English in origin (for example, Yiddish influence in the New York area; Black English influence). Languages in contact (as opposed to dialects in contact) rarely interact except in extremes and never in phonology. Rather there may be borrowings, which are eventually integrated into the target language (corduroy, diapers, etc.), or else one language obliterates the other, as in Latin taking over Gaulish, or English taking over Irish. However, the latter case is special, since traces of Irish remain in phonology, familiar syntax, and vocabulary. A standard form evolves when a group speaks such a mixture of dialects that no particular one can be distinguished.
The principles are the same for any language which `emigrates,' as with English, French, Spanish and Portuguese into North America. First the emigrant languages start to evolve from a specific homeland dialect. Then they evolve differently from the homeland because of lack of contact. The less contact, the more different the evolution, because the home dialect also continues to evolve.
The U.S. kept less of a `British' accent because it threw off British dominance at a rather early stage, which Australia and New Zealand did not. The educational system in those countries continued to import teachers and administrators from "home," as the British linguistic model was considered `superior'. Canada is an odd case: it is also British-dominated, but the Maritimes, which were the center of Canada for a significant while (from 1755 until around 1830),