Among the many catchphrases used by the Goons were a number of nonsense words – “spon”, “lurgi”, “yackabakakka” – which were probably introduced to demonstrate how daft the idea of a catchphrase was.
One regular was “ying-tong-iddle-i-po”, which was invariably immediately followed by one or more people shouting “GOOD!” for no obvious reason.
The catchphrase led to a song – ‘The Ying Tong Song’ – which gave the Goons their highest chart hit, reaching number 3 in September 1956.
The phrase reportedly emerged when Harry Secombe repeatedly mispronounced the name of Spike Milligan’s army friend Harry Edgington. Correcting Secombe’s “Edgerton”, Milligan apparently said “EdgINGTON, INGTON, YING-TON!” To which Secombe responded “iddle-i-po”.
The nearest thing I can get to a source for this is a poorly-copied version of an obituary for Edgington from a Goon Show message board, although Milligan mentions that Edgington was dubbed Edge-Ying-Tong in his war memoirs.
After serving as a gunner in the 19th Battery of the Royal Artillery alongside Spike, Edgington became a journalist after the war and later emigrated to New Zealand, where he died in 1993.
According to Wikipedia (citing Harry Secombe addressing a Goon Show Preservation Society event), Secombe was not allowed to sing on this record, or the other Goon records he appeared on, as he was signed as a singer to Philips Records. The Goons were signed to Decca, hence, Secombe only speaks his lines and it is Sellers and Milligan you hear actually singing (in character). Who the operatic tenor at the start is, I’ve no idea.