The person who put up the video didn't know it either.
Well, a blog-reader (James Buck) very kindly put us both out of our misery and told us it is 'Dear Almanzoer', written and recorded in 1927 by Oscar Celestin. He's right! How to watch the video? CLICK HERE.
Theme A (played once): 16-bar theme like a Verse of a Song. Typical of verses in Eb, it ends on the chord of Bb7, leading perfectly into the tonic at the start of Theme B.
Theme B (played once): 32-bar theme (like a Chorus of a song).
Theme C: a 12-bar blues theme (played 5 times: i. clarinet against offbeats - exactly as on the Celestin version, ii. clarinet against stop chords - again exactly as on the Celestin version, iii. cornet, iv. trombone leading, v. trombone leading - with decoration from cornet).
Theme B: the 32-bar theme is played ensemble again to finish (again as in the original Celestin performance).
Thanks as so often to the great video-maker digitalalexa for recording this performance for us.
Later, another very good YouTube video of this tune played by Tuba Skinny appeared. It was made by Dave Mann. You can enjoy it here: