Wikipedia sayeth, in part:

In The Oxford Companion to Music, Percy Scholes devotes about four pages to this subject, pointing out the similarities to an early plainsong melody, although the rhythm is very distinctly that of a galliard, and he gives examples of several such dance tunes that bear a striking resemblance to "God Save the King/Queen". Scholes quotes a keyboard piece by John Bull (1619) which has some similarities to the modern tune, depending on the placing of accidentals which at that time were unwritten in certain cases and left to the discretion of the player. He also points to several pieces by Henry Purcell, one of which includes the opening notes of the modern tune, set to the words "God Save The King". George Frideric Handel used the tune as the theme in the variation piece 'Sarabande' of his Suite No.4 in E minor, HWV 429, composed prior to 1720. Nineteenth century scholars and commentators mention the widespread belief that an old Scots carol, "Remember O Thou Man" was the source of the tune.