Greensleeves is an English folk song written and registered in London in the 1580s. Would somebody have been making money off of it? If so, who, and how?

The short answer is yes. You're probably referring to "A New Northern Dittye of the Lady Greene Sleeves," printed by a Richard Jones in 1580 at the Stationers' company in London. This is an example of a "broadside ballad," a type of printed music popular for centuries. Cheap single-sheet printings would be sold on the streets, each containing the lyrics of a new song. Sometimes, there would be an indication of a popular tune that the words could be sung to. Other times (and what seems to be the case here), the tune was newly composed and printed on the sheet alongside the words.

Since recorded music didn't exist, this was the 16th century equivalent of downloading the new Beyoncé single. You'd buy the songs that appealed to you, and then sing them yourself around the hearth or at the tavern. Many ballads were based off real events, especially the real tawdry stuff like executions, murders, love affairs, and great battles. Others, like this one, featured the sort of soapy drama that still reigns over pop today. "Greensleeves" is a song about one of the constants in life throughout the centuries: unrequited love.

Mr. Jones would have registered this new song, printed up a bunch of copies, and sold them to whoever wanted to buy them. Technically, he had a copyright and was the only person allowed to sell the song. But it wasn't so easy to keep a lid on things back then. First off, like the guys selling DVDs in sketchy parts of town today, there were plenty of people willing to sell unauthorized printings in order to make a quick buck.

Secondly, unlike today, the publisher wouldn't have gotten royalties from performances of the song. If I want to put "Blowin' in the Wind" on my latest album, I need to pay Bob Dylan. If I want to play it at a gig, the venue technically has to pay Bob Dylan as well (how strictly this is enforced varies by country and region). But back in the 1500s, if Joe Smith wanted to sing "Greensleeves" in the tavern with 100 of his closest friends, he didn't owe Mr. Jones a dime (or a schilling or whatever).

And to compound Mr. Jones' problems, since a lot of people couldn't read, they learned new songs aurally (from hearing them over and over), without the need for the printed broadside. So even if his song became the smash hit of the year, with everyone in London singing it in the streets, he might not see all that much profit. Still, it was obviously profitable enough that the practice continued through the 19th century, right up to the invention of recording technology.

EDIT: I wrote a little more about amateur music-making, mentioning the role of broadside ballads, in a comment a few months back: https://www.reddit.com/r/AskHistorians/comments/6htcpd/did_people_use_to_sing_more_in_everyday_life/dj18396/