As for Italy, near to impossible.

First, do you speak either of the languages? Totally fluent Italian is 100% needed.
In the Netherlands English is spoken almost as well as Dutch by virtually everyone. BUT you lucked out. To work in fields like education or nursing, you MUST speak well the language (friend I have in tou country told me it is actually mandatory).

Even if you can get past the language barrier and are fluent in either one, I would say that you are not really in the best position to get a placement. You need an offer for a job to get a visa but to get one is really hard.
In Italy you need an Italian/European certification for all public hospitals (the maiority since health care here is provided for free to all). In the private institutions, they still require an European certification and when they hire foreigners they hire people from East Europe, more likely to work for less money and trained the same way Europeans do. Also, they are required and do speak excellent Italian for what I have been able to see.

Only other option is to work as private nurse for the elderly. But the money involved is terrible and the precedence is given to Eastern Europeans anyway (there are agencies to get willing women from the East for what i understand. rather scammy businesses…) as it is easier to get visas for them (with the new EU countries).

With a local passport, things would not really be any better unfortunately. Now, if you were a male, things would be different (great lack of male nurses) but you are not in that cathegory either.
And I doubt the Netherlands would offer a lot more opportunities.
Usually people who succesfully get employed in Europe from other "rich" countries (and Australia classify as such) work in fields like finance, banking, marketing. I m not saying this is the only option, but to move an employer’s mind to actually hire you from Australia to work in nursing will not be easy since it never happens.
We are having some problems hiring fellow EU workes (due to some ignorance about how it should be done and how things like taxes should be handled), where the problem does not really exhist, imagine someone from another continent!

Some EU countries have "special lists" of occupations for which their is a need. Residence/work requirements are reduced. But if you take a job, you usually have to have a contract in hand and it has to be for a longer period of time, say 1-3 years. The rules are different in every country. I don’t know about the overlap with the working vacations visas. Sorry. You need to contact the immigraion bureaus in these countries, not the embassies. They don’t know much, as a rule.