Can we fly in Japan and/or Australia with EASA approved training and licences?
Generally speaking, you need to hold a licence issued by the same country that the aircraft is registered in. There may be exceptions but you cannot fly an Australian aircraft on your EASA licence. 

Luckily, converting a PPL between countries is generally straightforward when it has been issued by an ICAO-contracted state (which is most advanced countries and EASA).

For Australia, no additional tests are required (for PPL only, exams and flight tests are required for higher levels). You just need a medical, to pass a security check, and pay a fee. I believe EASA licences come with English Language Proficiency (ELP) level 4 to 6 so an English test is not required. Side note, the US does not have the ICAO ELP scheme so even native English speakers from there need to pass an English test when they convert to Australia. 

I'm less familiar with Japan but according to AOPA Japan you need to sit a law exam and obtain a radio operator certificate.