Games were televised and numbers were required on jerseys (but not necessarily 1 to 15).

From the "Old Collegians Rugby News", 4th April 1962

"The 'A' grade match on 5th May is to be televised.  Because of this all players have to wear numbered jerseys for games this year, as otherwise it is impossible for the commentators to pick out individual players.  The Club will obtain a supply of suitable numbers for distribution to players and these must be attached to their jerseys.  The purchase of the numbers in bulk by the Club will reduce the price at which they will be sold to individual members.

Because of the televising of matches, slovenly and incorrect dress will be rigorously discouraged by the S.A. Rugby Union.  Old Collegians Club jerseys (maroon and navy blue hoops) and socks are available at the Adelaide Sports Depot, Paringa Building, Hindley Street.  Members purchasing gear should be sure to ask for their Club discount.  The official dress is completed by navy blue shorts.  Any players having white shorts may wear them to training but they must not be worn for match play."

From the "Old Collegians Rugby News" of the 4th February 1979

" ... SARU has stopped us from numbering jumpers from 1-100 and each team must have numbers from 1-15."

In these early days of television everything was recorded "live" and no film was used.  Consequently, there is nothing in the ABC archives for any of these matches .... P.Horne, January 2002

However, the 1908 blue Wallaby jersey worn by Frank Bede-Smith, a centre who toured with the first Wallabies to the UK in that year, now in the possession of the Rugby archive in Sydney, features a Waratah emblem with the word "Australia" written underneath the Waratah and has a number five sewn into the back.
Assuming the number is contemporary with the jersey, the jersey proves that the Wallabies wore numbers in that era.  
Source: 'In Touch', Issue 3, 2001