Some confusion here.

It is true that ‘Kemps Jigge’ refers to William Kemp or Kempe (c.1560-c.1603), a member of William Shakespeare's (1564-1616) acting troupe. The tune that bears his name is not a jig as we understand it today: in Shakespeare's time it meant a lively farsical play after the main play. As well as playing major roles in Shakespeare's plays, Kemp's job was to perform these jigs, and it is probably from this that the tune gets its name: it was probably a tune that accompanied one of Kemp's farsical jigs. In 1612 the Middlesex justices of the peace made an order suppressing jigs because they provoked breaches of the peace!

William Kemp is most remembered for his energetic Morris dancing, which he once performed for nine days (twenty three days if you count the days off) during February 1600, dancing all the way from London to Norwich. This became known as his “nine dais wonder”, about which Kemp wrote a book, and for which the Mayor of Norwich gave him a forty shilling a year pension for the rest of his life. Since the tune that bears his name appears in the lute manuscript Dd.2.11, c. 1590-95, and his “nine dais wonder” was in 1600, then the tune cannot possibly be 'about' or 'in commenoration of' the feat, as many websites claim, because the tune was first, nor does he tune have any connection at all to the Elizabethan/Jacobean lutenist-composer John Dowland, as one website erroneously claims.

‘The Parlement’ and 'Nuttmigs and Ginger' are closely related tunes using essentially the same musical material. It is possible that the tunes are related in the same way that traditional families of tunes or songs develop: someone misremembered a bit, or decided they would rather play it a different way, and so it changes. However, the relationship may be only literary rather than traditional: Elizabethan composers, in the days before effective copyright, felt free to change melodies at will.

The Playford melody is musically unrelated and its origin is unknown.