"Weren't the Franciscan order monasteries important in England at the time of Henry VIII's expropriation of church properties? And weren't Franciscan monks known as 'blackbirds?'"
Wrong religious order. Benedictine monks with black habits would have been called Blackbirds. Franciscans wear brown, and they aren't monks, they are friars. Both were important in England in pre-reformation days.
I doubt if that's got much to do with the origin of the song though.