I had never come across the "29th of May" / "All Things Bright and Beautiful" link before.

In Grishkas link, The Fiddlers Companion says There is a "29th of May" in the music manuscript of John Winder (1789, Wyresdale, Lancashire), although the tune is that of "When the King Enjoys His Own Again." which is the one I knew. It then gives a list of references starting with Bacon (The Morris Ring), 1974. There doesn't seem to be a "When the King Enjoys His Own Again." in Bacon but there is "The Twenty-Ninth of May" -

X:1
T:Twenty-Ninth of May, The
M:C
L:1/8
S:Lionel Bacon - A Handbook of Morris Dancing - Headington
K:G
|:D3/2G/|B2B2B3/2c/d3/2B/|c2A2A2B3/2c/|d3/2e/d3/2B/ c3/2d/c3/2A/|B2G2G2:|
|:g3/2f/|e2d2d2G3/2A/|B2B2B2g3/2f/|e2d2c2B2|A3/2G/F3/2E/ D2D3/2G/|
B2B2B3/2c/d3/2B/|c2A2A2B3/2c/|d3/2e/d3/2B/ c3/2d/c3/2A/|B2G2G2:|

which I can't get to sound like either of the others.

Incidentally, my Grammar school, founded 1666, always used to celebrate Oak Apple Day on 29th May. Whether it still does since it became a comprehensive, I don't know.