A version of his tune is in Ceol Rince na hÉireann 3 under this name. I've never heard a name for it in "real life" or "reel life".
The Whistling Postman. First heard it on “the Maple Leaf” CD, with Jimmy Noonan on flute, Martin Meehan plays it on his CD “Three’s Company” and John Carty did it on “LAst Night’s Fun”. The eternal wonders of my iPod. Got even a live version played by Catherine McEvoy and Eileen O’Brien. And another live set [Reel of Mullinvat / Whistling Postman / Chattering Magpie] by Chris Droney Hmm. Tasty tune, nicely played Michael.
Thanks for the name.