It's been attributed to all sorts over the years. In 1859, Chappell discussed a series of claims (Popular Music of the Olden Time, pp. 691-707) dismissing some as untenable and leaving the question open, though rather inclining, it seems, to Dr John Bull rather than Henry Carey (later commentators have been less kind on the subject of Carey's claim, actually made some 50 years after his death by his son George, who was hoping for a state pension). Lully is dismissed, as the story came, not from a contemporary history, but from a much later novel purporting to be a mémoire.

The Oxford Companion to Music points out that the tune is a Galliard, and examines the way in which such pieces were typically constructed, together with various musical parallels in other pieces. It concludes, "It is probably a late seventeenth- to mid eighteenth-century recasting of folk-tune and plainsong elements... It seems as though there were certain phrases that drifted from tune to tune and found themselves a part of galliards during the sixteenth and early seventeenth centuries, of minuets during the later seventeenth century, of carols, and at last of a widely accepted patriotic song... If any attribution is necessary for song-books, the word 'traditional' seems to be the only one possible, or, perhaps, Traditional; earliest known version by John Bull, 1562-1628."

I should think that the current edition of Groves Dictionary would have as accurate a summary of the question as would be available.