From David Wulstan, Tudor Music (Dent, 1985, pp. 70-71) [Wulstan doesn't mention 'Greensleeves']:

Henry VIII's prowess as a composer has often been exaggerated, partly because of the romantic view of King Hal espoused by former historians of music, and partly because his subjects were doubtless obliged to be generous in their estimate of the king's abilities. The 'Service' which he is supposed to have composed for his chapel, mentioned by Peacham (1622, p. 99) no longer survives; if works of this kind ever existed, their musical worth is not likely to be much greater than those which survive in the manuscript in question [i.e., King Henry VIII's MS]. Leaving aside a few barely competent parts added to the work of previous composers, the king's musical vocabulary seems to have been decidedly limited. The famous 'Pastime with good company' starts off with exactly the same phrase as many of his other compositions, and in any case the tune seems to have been borrowed from a French theatrical chanson, 'De mon triste et desplaisir'. But although the musical interest of Henry's efforts is small, the words which he and the court composers set are an interesting stmosphere in his household during these years.

~Masato