Why does Australia have extensive uranium ore deposits?
An aptly named webpage from GeoScience Australia: Why Australia has so much uranium provides some insight as to some of the reasons behind the Australian continental relative enrichment in uranium.

Consider the following map from the GeoScience webpage:



One link that is stated on the website is that in regards to major uranium deposits" 


  observations support the conclusion that the large number of uranium deposits and prospects across Australia reflects the extensive emplacement of uranium-enriched felsic rocks in three main periods of igneous activity. While some uranium deposits appear to have formed during these widespread thermal events, most formed from uranium-enriched source rocks by subsequent low-temperature processes.


These 3 main periods occur:


Late Archaean (2.69-2.65 Ga)
the Palaeo-Mesoproterozoic. (~1.9-1.5 Ga)
and a bit younger in eastern deposits, the Silurian to the Permian (0.43-0.25 Ga)


The last point in bold above is elaborated further by the website, that uranium mineralisation being younger than the related intrusive igneous rocks.  Particularly evidenced by deposits found in calcrete, sandstone and unconformities:


  appear to have formed as a result of uranium mobilisation from older uranium-enriched source rocks under low-temperature oxidising conditions, and precipitation by redox reactions. The high-grade deposits are likely to reflect relatively more efficient oxidation-reduction systems.