Ordinary people in those days didn't know how to read and write spoken language, much less music notation. Although the words of many rhymes and both words and melodies of many folk songs were passed on by oral tradition we have no record of tunes to go with most nursery rhymes. Even those we now sing were usually set to music in the last century or two.
It seems likely to me that the ones we now call nursery rhymes were probably not sung at all, but called out in rhythmic speech, possibly accompanied by instruments like hand drums and tambourines. If so, they must have sounded a lot like what we now call Rap music. Even the meter is often similar. Can't you imagine "Humpty Dumpty sat on a wall" done as rap?
It's not likely ever to happen, but I'd love to hear a modern Rap group perform nursery rhymes.