Hi LizzieL
Not sure if I can fill in many of the blanks, but I can give you a hand.
Josiah Edwin Eltham was, biologically speaking, my maternal great grandfather. Legally however, he was my grandfather as he and his wife Alice (nee Oldham) adopted my mother who was Josiah's granddaughter. (Yeah, it took me years to get my head around that concept and I lived it!)
My mother (Josephine Ellen Bodin) was born in June of 1941 to Eric Bodin and Florence Bodin (nee Eltham, daughter of Josiah and Alice). Florence, alas, died three weeks later and Josiah and Alice adopted my mother upon which she was given the Eltham surname. (She married my father Denis Bland in 1961 and that was her final surname) For her entire life, she considered Josiah to be her father.
I do not know the exact circumstances of Josiah's arrival (alas my own mother died in 1995) but he was obviously resident here when my mother was born. I do know that prior to WWII he regularly traveled between England and Australia playing banjo on ships of the P&O line so shipping records would likely have him appearing regularly.
As a child, I spent many hours sitting at his feet listening to him play, and remember him a a tall smiling man of great humour, bearing and gentle nature.
Josiah died 15 August 1974 here in Adelaide, Australia; (Alice had died 7 September 1956) and Josiah, Alice and Florence share a grave in the Cheltenham cemetery here.
Josiah and Alice also had a son Joseph (my mother's uncle/brother), another professional musician (banjo again!), who died in '96 from memory. With Josiah, Joseph and Josephine, there was no shortage of 'Joe Eltham' in my life!
Not sure what more I can dredge up from the depths of memory, but I hope that adds another piece to your puzzle.
Cheers
David