Via a Yahoo! Answers enquiry, I just dug up some background on the Blue Peter theme (the earliest - 1958 - embedded above). A hornpipe-style piece called Barnacle Bill, it's a very popular fixture of the 'Proms' concerts (it was played 24 times in the decade 2000-201) and I'd always assumed it to be trad - but on reflection most of it is far too chromatic and "English light classical" in style. There is a trad song called Barnacle Bill, a bawdy American drinking song derived from a precursor called Bollocky Bill the Sailor - check out this clean version by Sourdough Slim. But no, the Blue Peter Barnacle Bill turns out to be a fairly modern composition by a Herbert Ashworth-Hope.
While the main motif is original to Ashworth-Hope, the introduction is clearly a quotation of the traditional tune variously known as The Sailor's Hornpipe, The College Hornpipe or Jack's the Lad. Any number of composers have similarly quoted it, such as William Walton in the Hornpipe segment of the 1923 Façade (see williamwalton.net for lyrics) and in the extremely catchy Hoagy Carmichael & Orchestra foxtrot version of Barnacle Bill the Sailor.
Ashworth-Hope (maybe 1880 - July 11th 1962) looks to have been an interesting polymath. Collating various music library and newspaper sources finds him to have been a successful solicitor and company chairman with several offices around the north-west of England. For part of his career he worked overseas with interests in tin and rubber; he was also one of the co-founders of the still-extant Gibb & Co. There's a photo of him - here, lower left - in Arnold Wright's Twentieth century impressions of British Malaya.
He ... returned to retire in Somerset in the 1930s to a home large enough to have a music-room that the BBC in the West used for broadcasts of concerts featuring the likes of Campoli and others. For the last years of his life he regularly watched Blue Peter marvelling perhaps that a harmless little hornpipe he had written in the 1930s would become as familiar to the nation as the National Anthem itself.
- Philip Lane, Naxos.com
Ashworth-Hope's home was the now-listed Marston Court in Marston Magna, Somerset. In fact he did anything but retire on his return to England, and he became chairman of Malayan Tin Dredging Limited and Southern Malayan Tin Dredging Limited, companies prominent in the 1930s-1950s. Barnacle Bill was just one piece in his small but solid corpus of light musical compositions: there are 25 on the Boosey & Hawkes list.