What are your sources for the earliest generations of this line?
I descend from Oliver Pocklington through his daughter Catherine (15 May 1656-18 Nov 1727) who married Walter Acton (Jul 1651-11 Mar 1717), second son of Sir Walter Acton, 2nd Bt.
Rev. Oliver Pocklington MD (1624-9 May 1681) was married to Catherine Towers (1627-1689) daughter of John Towers, Bishop of Peterborough (d. 10 Jan 1649) and his wife Mary_
I have never been able to trace the line past Robert Pocklington (father of John who d. 14 Nov 1642) and his wife Elizabeth_
I am certain that the dates you give for the earliest members of the line cannot be correct.
Some sources for you (can't find all of them at present):
Notes and Queries, Page 211, By Martim de Albuquerque, Published by Oxford University Press, 1857
“DR. JOHN POCKLINGTON.
(1st S. viii. 215.; ix. 247.; x.37.)
As several inquiries have been made… John Pocklington, D.D., Prebendary of Peterborough, Lincoln, and Windsor, and Chaplain to King Charles I., deprived by the Puritans, died 14 Nov. 1642, leaving issue, by Anne his wife, two sons, Oliver and John, and two daughters (Margaret, wife of Thomas Wright, 1653, and Elizabeth, living unmarried in 1642). His son, John Pocklington, is stated to have held lands at Higham Ferrers in Northants… Oliver Pocklington, Rector of Brinkton, co. Hunts, M.D., the other son of Dr. John Pocklington, died the 9th May, 1681, age 57: he left issue by his wife three sons, Oliver, William, and Charles, and one daughter, Catherine, born 1665, married to Walter Acton, citizen and goldsmith of London”
A History of the County of Bedford: Vol.3, By William Page (editor), Published 1912, Pages 175-179
"In the 17th century two remarkable men held the living of Yelden: John Pocklington, D.D., the High Churchman, and William Dell, the Antinomian. John Pocklington, Fellow of Sidney Sussex College, Cambridge, was presented to the living of Yelden in 1621 by Lady Say and Sele. He later became chaplain to Charles I. In 1639 he petitioned against Oliver Earl of Bolingbroke, the lord of the manor, 'for that "he hath wronged the church by enclosing and decaying the greater part of the best tillage" in Yelden,' and also sold a lease to one of his servants of a close worth £40 per annum, parcel of the glebe of the church. In 1640 John Pocklington appeared before the House of Lords on a charge of 'idolatry, superstition, and publishing pamphlets, wherein he defends all those innovations unhappily introduced into the church.' He was deprived of his preferments, and his pamphlets, Altare Christianum and Sunday no Sabbath, burnt by the public hangman."
“Dr. John Pocklington, (a prebendary of Peterborough, during the civil wars,) is supposed to have descended from a family who took their name from this town. He was chaplain to Charles I., and rendered himself so obnoxious to the puritans, by his two books, entitled 'Altare Christianum' and 'Sunday no Sabbath' that the Long Parliament deprived him of all his preferments, and ordered his two books to be burnt by the common hangman, at London, and the two Universities.”
I believe a Whig MP called John Pocklington was another son of Oliver, but can't find my source at the moment.