Currently doing an MSc in Systems biology at Warwick, but looking to relocate to Australia or New Zealand to do my PhD. I'm not especially fussed about the plant part, but it would be a bonus. I'm looking for a computing PhD, as I have an undergrad in Computation Biology, and like programming, so epidemiology of infectious diseases or population biology would be good. I'm not really sure how to search for PhDs in different countries, so any help on this matter would be great :-)

I think your best course of action is to search the Aus/NZ Universities for potential supervisors and then pop them an email! I'm a New Zealander and I have spent time at Otago University in Dunedin, NZ, and Australian National University in Canberra, Aus. I really don't know any specifics about your field of research (I study Developmental Genetics) but both of these Uni's are pretty strong in Biology at least.

If you're looking for funded PhD's I'm not actually sure where you could start with that. If you contact potential supervisors to register your interest they can probably tell you about relevant scholarships/PhD opportunities. Plenty of universities also have centralised PhD scholarships available (for any field) and these are often open to international students.

I guess I can also answer questions about NZ/Aus in general as well!