All of the countries you've listed require that you disclose your immigration history, including removals, refusals, and bans. Not doing so is a bad idea, and can result in visa denials, refusals, bans.
New Zealand is the same: you have to reveal your UK history and removal or risk not being issued a visa.
You noted on SETravel that you do not have a fiance in New Zealand but, rather, there is an individual who may be willing to marry. Culturally arranged marriage visas are issued by New Zealand.
New Zealand Immigration requires that you submit your passport (which may show the UK removal) and demonstrate your character (provide a Police Certificate from your home country, as well as from any country in which you lived for a year or more; that would include the UK).
You also have to prove that you've met (as well as her immigration status, among other things). Is she planning to meet you outside of New Zealand?
For your broad question, whether information is shared among countries such as the UK, Canada, the US, New Zealand, Australia, the answer is yes: all of these countries share intelligence information. No one knows what, or how much. Is immigration information shared? Who knows? Could be, but the bottom line is don't let that affect where you apply for a visa, or what you do or don't include in an application. Failing to disclose the facts, lying may result in visa refusals and can mean that you're not going anywhere for a long time.