Not according to my study. The origin you quote is one which is circulating widely on the internet. It seems that one site 'feeds' off the unreferenced 'information' on the other. The rhyme has often been thought to be a Jacobite slur on the Hanoverians, or to refer to William of Orange and his followers, but the first recorded mention of it is in 1675. I'm afraid I can't give you a reference for my view that the 'one in a velvet gown' is Charles II, because it comes from a pre-internet source, but if the first recorded example is 17th century then the context is just about right; it also fits the account of Charles' flight with his royal clothes hidden under 'rags'.

Granted that R J A White implies that it has Tudor origins, following a statute of 1563, but he in turn gives no reference to his assertion, and does not even try to explain the 'one in a velvet gown'. (WHITE R J A, 1967, "A Short History of England", Cambridge, The Cambridge University Press, p133). There is a reference to a different version of the rhyme in Ashley (ASHLEY W J, 1888, "An Introduction to English Economic History and Theory", London, Longmans, p352) which doesn't really help us much. The reference to William of Orange is supported by editions of the Oxford Dictionary of Nursery Rhymes (OPIE Iona & OPIE Peter [eds], 1951-1997, "Oxford Dictionary of Nursery Rhymes", Oxford, Oxford University Press), but Katherine Thomas suggests a Civil War origin with the Royalist siege of Bristol by Prince Rupert's army (THOMAS Katherine Elwes, 1930, "The Real Personages of Mother Goose", Boston, Lothrop Lee & Shepard) although Cullinan and Person throw doubt on her general accuracy (CULLINAN Bernice E & PERSON Diane G, 2001, "The Continuum Encyclopedia of Children's Literature", London, Continuum International Publishing Group). I can't find a reputable source that dates the rhyme to a time earlier than any of the above and, to my mind, the 'one in a velvet gown' line, whether an original line or one inserted to suit a particular set of circumstances, is a strong enough piece of evidence for that particular version referring to Charles II.