The following contradictory piece may throw some light on the situation.
From The Sunday Herald, June 23 1949.

London songwriters are mystified about an Australian song 'Suvla Bay', which has suddenly become the rage of Britain.

Sheet music copies credit both the melody and the lyric to 'Jack Spade'.

But Jack Spade cannot be found. There is some doubt whether he is even an Australian.

The BBC has made the song the hit tune of the month. Every 'pop' singer and dance band leader is asking 'Who is Jack Spade?'

The copywriters (sic) of the song are the Irwin-Music Company. They claim that 'the Jack of Spades' (a name often given in the profession to an unidentified composer) is alive and lives in England, but will give no other details.

Other London songwriters claim 'Suvla Bay' is a 1914-1918 tune 'pepped-up' for present day audiences. ('Suvla Bay' was the scene of some of the fiercest fighting in the Gallipoli campaign).

Mr Sonny Cox, director of Box and Cox, heard the lilting tune last year, secured the publication rights and reworded some of the Australian phrases which, he felt, were too strong for English digestion.

Last week he secured the English rights of 'The Road to Gundagai' and is now looking for more Australian tunes.

>>>>>>>This is either poor journalism or something is not quite right here. I got this from apparently a replica copy of the newspaper page online. Firstly it says they won't reveal the composer and then seem to be saying the writer is Sonny Cox himself. Secondly the dates don't quite fit. My copy of the sheet music gives copyright arrangement as 1944 and 1948. If it was recomposed in 1944 how can Cox have heard it only in 1948?. In my sheet music it is copyrighted to The Irwin Dash Music Co. Ltd., Publication rights for all countries except Australasia assigned Box and Cox (Publications) Ltd.

Incidentally the sheet music only gives one verse and the chorus.
The song was very popular in the RN just after the war but I think this was the sheet music version. The copy I have in front of me has Charlie Chester as the broadcaster but I have other copies somewhere.