Jim Adkins, who lives in the United States, has discovered a book called 'Adkins of Eastern Kentucky and Surrounding Areas' by Joseph T. Daniels, and published in 1988.

This book mainly records the families in the areas noted in the title. However, it also describes the possible origins of the Adkins name as well as a possible link to the UK for US families. As I have believed, this was in the 17th century, at the time when English traders were developing their business with the 'Colonies' - north America.
Here is an extract from Mr Daniels' book, firstly about the origins of the Adkins name:
    "In all probability, most, if not all, of these people are descendants of the Adkins Clan found in England as early as the 5th century. It appears likely that the name Adkins-Atkins-Adkinson-Atkinson-Ateykens, etc. was of Danish origin prior to the invasion and conquering by the Teutonics composed of the Angels, Saxons and Jutes.
.... [text omitted]
    It appears likely that most of the Adkins people that came to the colonies [North America] from England were direct descendants of John Adkins of Chard, Somerset, England. John was probably a descendant of William Adekyn of County Somersetshire [sic], born about 1327 and died about 1377."
Mr Daniels stresses that there is no proof of his theory but it must certainly obtain some credibility. Next he goes on to describe the family who he thinks started the Adkins family in the colonies:
    "In his will dated July 16, 1636, John the Elder, refers to 'my grandchild' son of John Adkins, who was born in Virginia, (between 1615 and 1636). In this will, John indicated that his wife was Katherine. His eldest son was Edward, with other sons Francis, William and John. These sons were living at the time the will was made. John also named his brothers, Thomas, Robert, William and Nicholas and his sister, Sarah Sellwood. He named his daughters, Hannah Ford, Elizabeth Smith and Katherine Wells.
    It appears reasonable that most of the Adkins clan are descendants of this John Adkins of Chard as he was a merchant and as such would be engaged in commerce with the Colonies. He had a son John and a grandson John that he referred to as being in the Colonies.
    There was a John Adkins that came to the Colonies (Virginia) before February 1623. In all probability, this was our John since John of Chard refers to his grandson, John, son of his son John that was born in Virginia in his will of July 1636. Several other 'Johns' emigrated to Virginia but these were around the middle of the seventeenth century, some few years after John of Chard's will was made and probated in the year 1636. This John Adkins emigrated from Bristol, England that is close proximity to Chard."
All very plausible. Remember that there would not have been many Adkins in Chard at that time, it was only a very small place then - and a place with a very interesting history!