I was born in the UK and hold dual citizenship UK/New Zealand and have both passports. I left when I was 4 years old and I am returning indefinitely in my late 40's. How long do I have to be living in the UK before I am considered a resident? Will I be eligible for health care as soon as I arrive? Can I apply for a UK drivers licence as soon as I arrive?
The main requirement for NHS entitlement is that you are a lawful and settled resident. You'll need your passport and some proof of address in the UK to register with a GP. There is no initial minimum period of residence, but the UK must be the country you spend most of the year in going forward. NHS may make further checks on your residence but will not delay any necessary treatment while doing so. Background info https://www.gov.uk/government/publications/guidance-on-overseas-visitors-hospital-charging-regulations
Since you are (also) a UK citizen, you can be considered settled as soon as you move to the UK with the intent of staying there. You should simply visit your local GP and register there, where they might ask you for your NI number (which you might not have, so you probably need to obtain it first), your passport and some kind of proof of address (obtaining this might take a while though, but not all GPs require it - especially from a UK passport holder). Once your done, you should be fine from that point on when you need to be admitted to a hospital.
Note if you are not a UK or EEA citizen, then even if you hold a long term visa you are not considered settled, and you need to pay the NHS surcharge to get access to NHS hospital services.
For the driver's licence, unfortunately you have to wait for 185 days after you arrive to be eligible to apply for one, or exchange it. Fortunately you can use your NZ licence for a year, so there would be an overlap.