This might be a case where two individuals have been conflated into one profile. And, according to one source, one line leads back to Edward I, so every connection along the way should be seriously scrutinized.
The Paul Abney, son of George Abney and Bathshua (last name unknown) was baptized on 14 January 1652/3 at St. Mary's in Leicester. Keeping with Church of England tradition at the time, it is likely that he was born shortly before that in early January 1652/3 or late 1652; however he could have been born years earlier.
This Paul Abney was part of the British Navy (possibly Merchant Navy; an expert needs to weigh in) not necessarily a pirate but that probably depends on one's perspective. He was the Lieutenant on the British Frigate, Josiah, serving under Captain Bartholomew Sharpe, sailing between Bermuda, the West Indies and other locations in the 1680s.
Some sources claim he was engaged in England to Mary Lee, the daughter of his mother's second husband, and arranged for her passage to Virginia with his brother, Dannett. She eventually married Dannett in 1702, so under this theory, Paul died sometime before that time but the place, location and any record of either is unknown. Further, it is possible that Paul never returned to Virginia to marry Mary Lee.
The source for this is:
Faris, David. Plantagenet Ancestry of Seventeenth-Century Colonists. Baltimore, MD, USA: Genealogical Publishing Co., 1996.
The book cites some original materials but not for every fact mentioned above. I believe the book has been updated since 1996. It is still under copyright and not widely available on the internet. The first edition is available on ancestry.com.
The naval records for Paul Abney's ship can be found here:
'America and West Indies: January 1686', in Calendar of State Papers Colonial, America and West Indies: Volume 12 1685-1688 and Addenda 1653-1687, ed. J W Fortescue (London, 1899), pp. 135-147. British History Online http://www.british-history.ac.uk/cal-state-papers/colonial/america-west-indies/vol12/pp135-147 [accessed 25 July 2017].