I've been looking for an online version of the melody for this song. I found this at the National Library of Australia:

Pergolesi, Giovanni Battista, & Moore, Thomas. & Wales, Evelyn, & Pergolesi, Giovanni Battista, 1936 When love is kind [music] : old Austrian air / words by Thomas Moore ; arr. by Evelyn Wales ; Nina : canzonetta / G. B. Pergolesi ; English version by Evelyn Wales

At first I thought that Pergolesi had written the tune for Moore's song. Now I believe that the library simply owns a document that had two entirely different pieces of music in it, one by Pergolesi and one an Austrian tune. (I say this to keep anybody from spending more time on Pergolesi, which would probably be wasted.)

There are a number of YouTube videos of high schoolers singing the song, so somebody out there has the tune. However, it would be SO MUCH effort to get the melody from those videos. For one thing, the piano is too loud.

So if anybody knows where to find the tune....