Some biographies I’ve read about Hal mention this and then state it’s probably apocryphal without going into specifics. Does anybody know why the songwriting credit was attributed to him once but is not now?
It’s a good old-fashioned urban myth. Although he was certainly a keen musician, there’s no evidence whatsoever that he wrote the tune.
I think Greensleeves predates Henry VIII. He gets credit for the song “Pastime with good company”, though.
A more detailed version of the ‘myth’ is that he wrote it for Anne Boleyn, before she became his second wife, she had (or was said to have - no cite but I was taught this in school) six fingers on one of her hands - understandably, in an age when physical abnormalities were often associated with witchcraft or evil doing, she was sensitive about this and had her dressmakers design wide sleeves which hung down to obscure the hands - hence Greensleeves refers to her 
musical composition was not an unusual ability for a gentleman in those days - it was an accomplishment, not a profession.
kind of a cafe society-style hijack, but flanders & swann’s take on how greensleeves came to be written (involving Henry VIII, yes,) is just hilarious. The number is on ‘At the drop of a hat’, [lyrics here, but reading them doesn’t do it justice.] I don’t really understand all of the in-jokes and references to old english culture, but I can’t stop laughing when I hear it. 
The six-fngers myth came into play after Anne was long dead. The sone of Thomas Wyatt, one of the men who courted her, wrote that she had a small nub of fless on the side of her finger with a bit of nail on it. However, he never met Anne.
During her lifetime, even heravowed enemies at court did not write about any deformities.(They called her the “Goggle-Eyed Whore”.) A lady in waiting who had met her in the French court claimed she had a “wen” on the side of her neck and a sixth finger, but no one who was around her at Henry’s court said anything of the kind. And they would have-- Anne was a woman with a lot of enemies.
In court, it would have been impossible to hide a deformity. Ladies in waiting bathed, groomed and dressed Anne from the skin out. Bribery of ladies in waiting was extremely common, and not all of Anne’s ladies were her friends. There’s no way she could have kept a deformity a secret. All of the court would have known, and they would have mocked her for it.
Secondly, Henry was a very superstitious and paranoid man. Desperate for a son, he would not have wanted to marry a woman who had deformities, which were considered a sign of association with the Devil. Up until the last year or so of his seven-year-long divorce, he was still pretending Anne was just an innocent maiden in the court, not his intended bride. He would have had a much harder sell if it were known his beloved was marked by Satan.