Register of The University of Oxford, A. Clark, ed., Vol. II, Clarendon Press: Oxford, 1887.

part i, p. 306
2 Oct 1592; Hugh Buckner signed an oath as bailiff of the city of Oxford.
part i, p. 319
1596; Hugh Buckner signed a petition to claim £20 from Richard Smout, a bankrupt carrier of Oxford.
part ii, p. 306
12 Oct 1621; William Buckner matriculated at Christ Church College. From London, son of a gentleman, 16 yrs. of age.
part iii, p. 422
Degrees conferred on William Buckner at Christ Church College: adm. B.A. 21 Oct 1623, det. 1623/4; lic. M.A. 13 Jun 1626, inc. 1626.
part iii, p. 332
Degrees conferred on Thomas Buckner at Magdalene College: adm. B.A. 15 Dec 1614, det. 1614/5; lic. M.A. 18 Jun 1617, inc. 1617; lic. D.D. 15 Mar 1638/9, inc. 1639. <Demy of Magdalene Coll. 1610-1618, Fellow 1618-1631, see Bloxham's _Register of S. Mary Magdalene College_, v. 5, p. 41. >

Notes:

The notation that Thomas was a demy of Magdalene College from 1610 to 1618 gives useful information about his age. The position could only be held from the age of 12 to the age of 25, though in later times the upper restriction was seldom enforced. Thus we see that Thomas had to have been born in or before 1598 (1610-12) and probably in or after 1593 (1618-25), not quite as exact as we get from William Buckner's matriculation data, but pretty good.

Thomas Buckner was kind enough to leave a will, which fairly certainly identifies him as the son of John Buckner of Abingdon (b. ca. 1565) and the grandson of Thomas Buckner of Whitley, Berks. Thomas' christening was recorded as July 17th, 1593 and the will indicates he was born on June 24th, which perfectly accords with the university matriculation records.

William Buckner is also fairly identifiable, since the only Buckner in London who would have been described as a gentleman when he was born around 1605 was Thomas Buckner the mercer. In fact, there is a christening record of a William Buckner, son of Thomas Buckner, dated Oct 27th, 1605 in the parish of St. Bartholomew at the Exchange, so the identification seems certain. William had two children in Salisbury with his wife Elizabeth, Elizabeth and Anthony, but it is not clear if either of them survived infancy.

Hugh Buckner, the bailiff, is the least easily identified of the three, but I think it's fairly likely that he was the son of John Buckner of Botley, Berks. (d. 1609) and the grandson of William Buckner of Botley (in Cumnor parish) (d. 1558). He was thus perhaps the nephew of Rose (Buckner) Smith who married Thomas Smith, a mid 16th century mayor of Oxford.