I have never seen anyone actually establish any of the "rose" rounds with much authority or antiquity (I learned them in Girl Scouts, too). The melody and lyrics for "Hey ho, nobody at home" appears in Ravenscroft's Pammelia (1609), no. 85, but the last bit is not the modern familiar one:

Hey ho,
Nobody at home
Meate nor drinke nor
Money haue I none,
Fill the pot Eadie

It's marked for 5 voices; the melody sung nowadays makes six parts.

How and at what point somebody decided to turn a snappy drinking song into a dirge I haven't found. There is a catch in Apollonian Harmony (ca. 1790) called "Ding dong boam bell" all about funeral bells that has some melodic similarity.