Cyber Hymnal has three tunes listed for "All Things Bright and Beautiful." The one they (annoyingly) play in the background is ROYAL OAK, by Martin F. Shaw, so I suppose they deem that the most common tune (and I'd agree). The second tune they give is titled BRIGHT AND BEAUTIFUL, by William H. Monk. the third tune is GERALD, by Ludwig Spohr, 1834.
Cyber Hymnal says the lyrics are by Cecil F. Alexander, published in 1848.

This page has the melody for "The Twenty-Ninth of May." I can see how that would fit "All Things Bright and Beautiful" very well.

But this plethora of melodies has made it impossible for me to learn this song. Every time I try to sing it, I switch melodies midstream. Oh, the agony of it!

-Joe-