Eric Partridge agrees with the OED that the musical "busk" originally meant "peddle". The change from "playing in the street for money" to "playing in the street for any purpose" is a small one. (This word is ultimately from Spanish buscar and Italianbuscare, "to seek", which in turn may be from Germanic words related to "bush"--as in "beat the bushes for something".)

"Busk" in that Scots song is unrelated. It means "prepare oneself, get ready" or "hurry", either of which makes sense in the song. (It's from an Old Norse word for "prepare oneself", and may be distantly related to "bound" as in "bound for glory".)

Now pretend you're interested, Alex!