Hello!
I made the same move as you around 10 yrs ago with a 3.5 yr old and a 11 month old. I am the NZ er and my partner is from the UK.
We airbnb'ed and then rented near Devonport and lived in that area for 2 yrs. We found the move pretty easy with young kids. We moved in Nov and it was really good time of yr to move as it's warm and if all of your furniture as not arrived it's much easier in summer.
We ended up moving to West Auck when it was time to buy a house. If thats your goal too, my advice would be to move to an area you can afford to get on the ladder as making new friends all over again was a drag. Devonport/The Shore is lovely but so pricey.
In terms of childcare... there are 3 or 4 main options
In-home care: Barnardos and Porse offer this. Usually v affordable and some amazing people working in this space.
Commercial nursery: as you would expect based on uk experience, but probably a bit more laid back, and cheaper.
Church/charity nursery: there are often nurseries attached to churches or community associations. They are not necessarily religious, but offer this service.
Auckland Kindergarten Association: my kids all went to Kindy. This will give you a real NZ pre school experience. Quite long wait lists but cheap as chips and great education.
Playcentre NZ: this is not childcare but a place you and your children can join and play and grow together. I cannot recommend this enough. This is a recommendation for YOU and your children. This is a place that kids and parents attend together. You will make lovely friends or just acquaintances who you will then see around for yrs as your kids grow up together. you can do 1 or 2 sessions a week, and use other daycare providers as well.
Overall my advice is not to worry about the nursery settling too much. I think you'll find it a very gentle experience that you can take at your own pace.
Happy to answer any other questions.