Do white Australians have a distinct look?
Yes, it is possible (on some average basis, i.e. with chance better than random).

Take a look at average women faces from 41 different countries (AFAIK created with Face Research software; sadly no picture for Australia). Clearly, even from small and nearby European countries you can spot differences (which work on the average). I guess the case of Austriala vs other countries isn't an exception. Similar face averages are also on the project Face of Tomorrow. 

However, when dealing with a single subjects it it may be not a reliable method to tell of his/her nationality or country of origin, as:


even within an relatively isolated group there is variance in genotype and phenotype,
people do move (especially now) and do mix (so unless you are comparing isolated islands you should't expect to have homogenous genotype).


Moreover, when identifying a person we may (even subconsciously) look at different features (including clothes, accent, behaviour, context, ...).