Yankee twang: In remote corners of East Anglia today, country folk still speak in a harsh, high-pitched, nasal accent unkindly called the "Norfolk whine." This dialect is the survivor of a family of accents that were heard throughout the east of England in the seventeenth century, from the fens of east Lincolnshire to the coast of Kent.In the Puritan great migration, these English speech ways were carried to Massachusetts, where they mixed with one another and merged with other elements. During the seventeenth century they spread rapidly throughout New England, and became the basis of a new regional accent called the Yankee twang.(1)
Southern drawl: The speech ways of Virginia were not invented on America. They derived from a family of regional dialects that have been spoken throughout the sough and west of England during the seventeenth century. Virtually all peculiarities of grammar, syntax, vocabulary and pronunciation which have been noted as typical of Virginia were recorded in the English counties of Sussex, Surrey , Hampshire, Dorset, Wiltshire, Somerset, Oxford, Gloucester, Warwick or Worcester.(1)
(1)Albion's Seed,David Hackett Fischer
Yes, the source of the American accent is based on various regional accents in England, Scotland, and Northern Ireland. Some of those regional elements still exist in the UK but they are not “BBC English”, so people don’t think of them as an “English accent”. What’s also interesting is that using “aks”/”ax” for “ask” can be found in Old English as well as Chaucer’s Middle English (it can be found in The Canterbury Tales).