No agreement on pomegranate-immigrant-pommy.
The earliest use of the term occurs during the 1st World War, but not always meaning the same thing (Now generally applied to an immigrant from the United Kingdom).

1915, Gammage, in Broken Years- "We call the Regulars- Indians and Australians- 'British'- but pommies are nondescript."
1916, ibid- "They're only a bastard lot of Pommie Jackeroos and just as hopeless."
1916- Anzac Book- "A Pommy can't go wrong out there if he isn't too lazy to work."
Pom
1919, Downing, "Digger Dialect"- "an English soldier."

The pomegranate explanation didn't appear until 1923, in D. H. Lawrence, "Kangaroo."

I'd say it's still open to argument. (Above quotations from the OED).
See W. S. Ramson, "Australian English."