If the second known version (1611, in Melismata, also reprinted in Chappell) were the oldest, this might be possible — there are seeming political references to "Gib, our cat" and "Dick, our Drake." But the Wedderburn text, which at least anticipates the song, predates the reign of Queen Elizabeth by nine years, and Queen Mary by four.
If it refers to any queen at all, it would seemingly have to be Mary Stuart. Wells, however, in the liner notes to the LP Brave Boys; New England traditions in folk music (New World Records 239, 1977), suggests that the original may have been satirically altered in 1580 when it was recorded in the register of the London Company of Stationers, as this would have been at the height of the unpopular courtship.
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Some versions end with a cat, snake or other creature devouring the couple and wedding guests.