The two prominent Puritan migrants, John and Ananias, were found to have lived in Nottingham as well as Kingswinford, Staffordshire. Ananias arrived in Salem, MA around 1638 & followed by John a year later - which is the basic story in "The Conklings in America" - see page 17.
Lawrence H. Conklin's Geneology Essay has a guess that their family were probably French Huguenots who emigrated to England near 1570 and John & Ananias were probably the second generation in England. While 16th century English spellers weren't super consistent this is an explanation for the wide variation in spellings of the surname.
There's also a third Conklin (Jacob) who emigrated to Salem that no one can figure out the parentage, but he also seems to have been from the Midlands, I think the consensus is John and Ananias were brothers and Jacob was a close cousin. A Jacob Conklin married and died in Salem and his children appear to have been adopted by Ananias which causes some confusion with a fourth group who might be Jacobs children or another family altogether. No one can quite figure out the parentage for any of these Conklins. Hope this helps someone!