The question of whether private school decisions are amenable to judicial review has yet to be answered in New Zealand.
James Anson-Holland's recent article The Amenability of Private School Decisions to Judicial Review published in the Public Law Review (2019) 30 PLR 106 discusses the conflicting judicial and academic debate within the Commonwealth in order to conclude that private education is sufficiently public to attract judicial review.
The article goes on to suggest that, even if a court were to decide private school decisions were not amenable to judicial review, the contractual context (between school and student) justifies equivalent procedural protections as an implied term.
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