If we were to be married in New Zealand (a country that allows same sex marriage) can we then migrate to London where he would be able to live and work as my spouse?
There are a number of issues here, and it's hard to know which you're asking about, so I will touch on a few of them.
First, as you may know, the complicated history between Great Britain and Ireland has resulted in a situation in which Irish citizens can readily settle in the United Kingdom. Furthermore, Ireland and the United Kingdom are both members of the European Union, which means that EU directive 2004/38/EC, establishing a right of freedom of movement within the EU, will apply to your case. If the UK leaves the EU, of course, the situation may change.
The full consolidated text of the directive is at http://eur-lex.europa.eu/legal-content/EN/TXT/?uri=CELEX:02004L0038-20110616.
Directive 2004/38/EC actually concerns "the right of citizens of the Union and their family members to move and reside freely within the territory of the Member States" (emphasis added). In fact, while it defines "family member" to include spouses and registered partners, it also extends free movement rights to "the partner with whom the Union citizen has a durable relationship, duly attested."
It therefore may be unnecessary for you to get married in New Zealand as a prerequisite to moving together to London. Assuming your relationship qualifies as "durable," you could move together to London and then marry there (but not in Northern Ireland), or not marry at all, and your boyfriend would enjoy the same right to live with you and to take employment.
On the one hand, the burden of proof is somewhat higher for unmarried partners, but on the other hand, if you apply for freedom of movement rights soon after your marriage, you may be suspected of a marriage of convenience, in which case your burden of proof will be similar: you'll have to show that you've been living together as a couple for some time.
The UK has developed the EEA family permit to serve as entry clearance for non-EU family members of EU/EEA/Swiss citizens. Your boyfriend doesn't need one of these to enter the UK because, traveling on his Australian passport, he will not need a visa. The family permit is free of charge, however, and can be useful: it gives you an opportunity to document your relationship officially while you are still in Australia.
Once you move to London, your boyfriend will need to get a residence card. The residence card is optional if you are married or in a registered partnership, but required if you remain unregistered partners.
(Even if you're married, it would be a good idea to get the card, because without it your then-husband would have to show your passport and his/your marriage license every time he wanted to assert his 2004/38/EC rights, for example when applying for work or traveling with you in the EU.)
The UK.gov pages for the EEA family permit and the residence card include information about the documents you need to submit and so on.
The residence card will be endorsed "family member of an EEA national," making it an Article 10 residence card (after artice 10 of directive 2004/38/EC). EU border officers, or at least Schengen border officers, are not supposed to stamp the passport of someone traveling with such a card.
Another little-known benefit of the right of free movement for non-EEA family members is that they are entitled to use the "EU passports" lines at Schengen border posts. This is explicit in the Schengen Borders Code. This is true even if they do not have a residence card, as long as they are traveling with or joining their EU family member.
(In practice, using the EU line when joining the EU family member would probably cause more trouble than it would be worth, since you'd have to prove the relationship as well as the location of the EU family member.)