No; it isn't old enough for that. It's quite easy to consult the Opie's Oxford Dictionary of Nursery Rhymes (available in all good libraries), and it's a constant puzzle to me why people rarely bother to do so, preferring instead to repeat long-discredited Old Wives' Tales. First printed in Britain in 1881, but apparantly known in Massachusetts around 1790; where it had none of the later accretions which people nowadays imagine to refer to the Black Death.