If you hold an Oz Class 1 then you should have no worries getting a US Class 1. Bear in mind that in lots of commercial operations in the US only a Class 2 medical is required, unlike Oz (Class 3 for private ops).
A JAR Class 1 could be problematic. Some of the requirements are more stringent than Oz. There are limits on the maximum vision correction allowed (less correction is allowable for an initial issue, a bit more correction for renewals). As I recall there are other investigations that have to be done that I don't recall being part of the Oz Class 1.
NB: Costs about US$80 - 100 for a US Class 1 and several hundreds of pounds for a JAR Class 1 (in the UK).