As you'll be able to tell from my name, I'm from NZ.
I've just done the opposite, come from NZ to the UK. It's hard. Damned hard. I don't know how old your kids are, but mine were 12 and 14, and it's very hard on them, leaving their extended family (even if not particularly close, which we weren't to most of ours, but it means they get even more distant if you know what I mean), their friends, their school, the culture and way of life (which are different), and the school system which they know and you know.
We did it because husband's work put a lot of pressure on him to relocate, and made an offer we couldn't refuse. We got a very good relocation package to cushion the cost to us. Even then it cost a lot.
The homesickness, even if you're willing, and the guilt, can be fairly crushing at times. It's hard when things happen, and you're not there. I feel a bit like I'm neither fish nor fowl - neither Kiwi nor Brit!
For me, emigration is a bit like having a baby, it takes two yeses to make it happen, and only one for it to be a no goer. It is VERY expensive, no matter which way you go.
A few points to consider about New Zealand:
It is not a tropical climate. We do get a lot of rain, wind, and cold. The further south you go, the colder and wetter it gets.
It is not a high wage economy. There is a very high cost of living. Particularly in Auckland. Power costs are high, insurance is high, cost of white ware, cars etc, is high, as is food.
Housing is expensive. It is different to the UK (Please note I say different, not necessarily better or worse). It is not as well insulated, particularly in older properties, older properties are less likely to have double glazing, and central heating is rare. On the plus side, they tend to be bigger, and more modern, open design, particularly if you buy newer. But, as I said - expensive!
Education system is quite different. Personally my kids feel the depth and breadth of education they are receiving at a comp' in the UK is superior to what they were getting in NZ. (Then again, they're academic, and love a challenge).
You fellas over here see a lot more open to newcomers than NZers are. Once again, I think the further South you go, the more likely we are to be courteous, friendly, but not necessarily fold you in. If you're into sports, clubs, and your kids are as well, I think it's a lot easier.
Your concerns about your family ties and obligations are valid ones, and your partner should respect that. The fact that he is not as close to his family is no excuse. I have a lot of guilt because my elderly, widowed mother (in good health thankfully) is now the responsibility of my sister and niece - although they know I do what I can in terms of maintaining contacts, regularly talking, etc, and will always come home if she goes pear shaped.
I really think it is piss poor your partner's attempts to emotionally blackmail you, with saying that you will have to go fulltime, so he can go part-time. When it comes to that sort of thing, surely it's about what's best for the kids at this point. Does he want to go part-time for him, or because he wants to spend time with the kids and enhance the family quality of life and day to day experience? I agree that you either need to have some good solid talks about the underlying issues, or some couples therapy to sort out the underlying issues.
Sorry for the novel.