Claude M Simpson, The British Broadside Ballad and Its Music, 495-6, makes no guesses as to authorship, but says:
"The title of this dance tune recalls to us the old custom of naming a king's younger brother 'Monsieur.' The reference here is evidently to François, Duke of Anjou, and brother of Henri III of France. It may be assumed that the tune was in circulation before François's death in 1584. A Continental version of 'Allemande Monsieur,' c. 1600, is preserved in Het Luitboek van Thysius, No. 311 [tune quoted]; English settings include 'mounsers almane' in the Wickhambrook Lute MS, c. 1600, fol. 17, and 'Mounsieurs Almane' in the Welde Lute MS, c. 1600, fol. 14v."
He refers to other examples; Byrd's arrangement of 1591 and Morley's of 1599 among others.