Barking
Sorry, only just seen this.
Yes it is possible but would be quite a long process.
First of all you'd get a study visa which enables you to work for up to 20 hours per week if you are doing a course of more than 2 years duration. This should be relatively straightforward, no points required, just need to be accepted to study by a recognised institition and willing to pay the fees. Your other half would have to apply for a visitor visa though (not entitled to work in theory although you might be able to get around this if he was still working in the UK but based out of NZ).www.immigration.govt.nz/migrant/stream/study/canistudyinnewzealand/
At the end of the study period you should then fairly easily be able to get a work visa for 1-2 years. There is a Study to Work scheme as well but this requires you to study for a qualification in one of the skill shortage areas, e.g. teaching as Sibble suggests.
If you could get the Work Visa extended out to 4 years (should be possible in the right job with a supportive employer under the Work to Residence scheme) you could then apply for residency!
Have you tried working out whether you or your DH would have enough points now to apply under the Skilled Migrant programme? Strangely enough because you are younger you might find you "score" more points than your DH despite not having been in the workforce recently. The points system is for this scheme only - work or study permits do not use points.
You get an extra point for saying you will settle outside of Auckland - might be the tipping point!
Click on the "Points Check" link here www.immigration.govt.nz/migrant/stream/work/skilledmigrant/ to see whether you'd score enough points to be AUTOMATICALLY eligible (subject to health/crime etc checks). If you're only a few (2-3) points short it would still be worth applying as you'd go into a pool from which they select people throughout the year to make up the Minister's specified quota for the year.
Skill shortages tend to focus on technical skills plus the entire medical profession. have a look www.immigration.govt.nz/nzopportunities/opportunities/
One option worth looking into would be to see which of you scores the most points and then for the other to look at getting a study permit.
I'm not sure but if you are here under a work visa (but not a study visa) I think you should be entitled to free state education for your kids - unless you want to go private (and your decision on that will probably depend on the quality of the local schools where you settle, some are great, and particularly at primary level you should be fine in a state school). This is the government website where you find the equivalent of Ofsted reports for all schools including private schools. www.ero.govt.nz Under a study visa your kids would probably also need to apply for study visas and then you'd need to find a school that takes international school students. This shouldn't be too hard in a bigger city like Wellington or Christchurch.
Having a sponsor won't make any difference to your points unless they are close family (extra points) or willing to sponsor you for an actual job (which they must demostrate they can't fill locally). However they may be helpful in enabling you to get a study permit.
Pretty much anyone can open a bank account. Its much easier than in the UK. They'll even give you a Switch type card straightaway!!
As a UK citizen you have automatic recipricol rights to public health access for emergency (A&E) care regardless of which permit you're on. If you have a work permit you should have the same rights as NZ residents. The tax system is different here, public health costs come out of general taxation, there is no separate levy per se. Having said that probably about 25% of the population have private health insurance to avoid waiting lists for elective surgery and specialist treatment. Indicatively that costs me about $70/month for 1 adult and 2 kids.
I hope this helps! BTW I'm a kiwi living in Wellington, and currently on maternity leave.