In the world of childrens' TV over here in the US, one example I can think of is Edward White's sprightly ditty "Puffin' Billy" from Chappell, repurposed for many years as the theme to the "Captain Kangaroo" show.
On a more local scale, the local Chicago series "Garfield Goose and His Friends" used one of Robert Farnon's "Sounding Brass" tracks for Chappell as an opening fanfare. The main theme itself ("Monkey on a String", performed by organist Ethel Smith) isn't library, however, although the composer (Denes Agay) composed some ditties for Sam Fox.