As long as you have paid, your Domain Name will be registered straight away. You will get an email with the subject line "Domain Registration Confirmation"
.NZ domains won't start working until on the hour as that is when the .NZ Zone is updated. .com/.net/.org domains should start working within a couple minutes as long as the domain is setup in your Web Hosting Account or you have just ordered one.
All domains will propagate within 24 hours.