Janice
The medal roll for the Southern Rhodesia Volunteers, Western Division, (WO100/267 page 22) shows "508 Tpr. Edwin Allen Atkinson" was entitled to clasps "Relief of Mafeking", "Rhodesia" and "Transvaal" on his QSA medal. There is no marginal notation concerning him on the medal roll page. The Kaplan roll for the RofM clasp shows him as serving with E Cyclist Troop of the SRV.
I have never seen any reference to the recruiting of volunteers in NSW (or elsewhere in OZ) for Rhodesia - have you considered looking at the newspaper index TROVE? There may be shipping lists available as well. Rather then being recruited for the SRV in Sydney, NSW, it is more likely that Edwin Allen Atkinson travelled to Rhodesia under his own steam. Most likely pre-war.
Unfortunately, researching men with service in the Rhodesian units can be very difficult. There is one reasonably useful source; a pair of books titled "Rhodesia Served the Queen" - however they are very scarce. You might be able to read them on-line? In Vol. 1 there is some detail of the formation of the various components of the SRV - but no specific reference of your Atkinson.
Regards
IL.