I can't see any attribution for the tune earlier than Waller.

It doesn't sound at all like a harp exercise, and doesn't have a lot of arpeggios. It sounds a lot like Schubert, Schumann or Brahms, and somebody of Waller's social background would have been much more likely to listen to them than to either Irish traditional music or Celtic-twilight revivalist work.

Compare it with Brahms's well known lullaby op.49 no.4; not the same tune but very similar.