I recently played some three part compositions attributed to Henry. They were all excellent and very definitely in the style of his era. Even if he had help in writing some of them it goes to show some compositional tendencies. Also, the simplicity of Greensleeves when compared to these mich more intricate works suggests that Henry could well have written it. Remember, it doesn't take a musical genius to write a catchy tune.
I would also like it to be remembered that extensive musical education was part of the upbringing of such people as much as Latin, Greek, arithmetic, etc.