Many online family trees show that William Adkins of Henrico County was the son of John Adkins of Sandy, Bedfordshire, England, and his wife Elizabeth Bromwell.
Unfortunately, this has been proven false. The facts from the Bedfordshire Parish Register are (https://archive.org/details/bedfordshirepari06bedf/page/n5/mode/2up):
John Adkins was baptized on 7 Apr 1639, (found on page A16)
Elizabeth Bromsall (not Bromwell) was baptized on 21 Dec 1642, (page A17)
John married Elizabeth on 4 June 1665, (page A65)
Their son, William, was baptized on 28 Mar 1689, (page A26)
John was buried 17 Dec 1692, (page A107)
Elizabeth, listed as a widow, was buried 27 May 1699, (page A108)
William, listed as an orphan, was buried 9 Sep 1704, (page A109)
Other theories for William's parentage include Thomas Adkins, Jr. of Henrico County, VA and John Adkins of New Kent County although neither can be proven or disproven at the current time.