What is New Zealand like (weather wise) around the end May and first week in June?

Starting to get cold weather is okay.

NZ is completely enclosed within a dome and as such we do not have weather. We do have a climate. The warmth is created from the hot air from American tourists who have plenty to say but don't have time to listen and learn. The cold is from the icey stares we give Nth American grating nasal accents (in my experience the worst in the world.)