The Radio 4 speaker was probably an idiot. There are very few songs with such a definite date as Greensleeves - 1580, a generation after Henry VIII was dead and more than 100 years after the end of the Middle Ages on anybody's count. Both words and music are unmistakably of that time and no earlier.

There are several good reasons for not lying when you present your music as being mediæval. The one that matters to me is that the music is the soundworld of a particular time - it gives you an insight into how people felt and thought back then; what kinds of art moved them, and why. You can find the original words for Greensleeves on Mudcat: Bruce Olson posted them. He thought they were crap and said so - most of us would agree. A mediæval listener would also have thought they were crap, but maybe for different reasons. Compare it to a genuinely mediæval song and you can see what had been lost by then.