It seems to be widely believed that Winston Churchill had a hand in it. He had been very supportive of Edward and there's more than a hint of Churchill's turn of phrase.
I don't know whether that's ever been actually confirmed though.
It may have been Churchill, but who cares, really! We should all be grateful that he decided to abdicate! A spoiled brat, a sycophant of Hitler and his Nazis in our extremity of danger, according to a British TV program, he gave away as many secrets as he could - it was just as well that he wasn't trusted with much that was important.
Without having the necessary history books to hand, I'm guessing that it would have been drafted, at least, by the King's Private Secretary with the assistance, probably, of the Prime Minister of the time (who was Stanley Baldwin). Churchill seems unlikely since he was out of favour with the establishment at that time.
Philip Ziegler wrote the definitive biography of Edward VIII so that would probably be a reliable source.