IIRC HM the Queen also uses a chartered British Airways airliner for State/Royal visits.
The US situation with Air Force One is a bit different, in that Air Force One also has some abilities as a flying command post/"Whitehouse". Given the US nuclear arsenal and the past (and unfortunately again, rising) potential need for their deployment, the POTUS needs to have instant communications if/when needed. I would be interested in what the British have in place for such a situation. Does the decision to launch nuclear weaponry rest with the PM, or the Sovereign? But I digress...
As for the costs of operating the A400M vs. C-17... Respectfully, until the A400M has been in service for several years, I doubt that any real understanding of the operating costs will exist. Even then, unless/until more A400M users exist in regions closer to NZ, A400M operating and maintenance costs could be higher than currently estimated. With the US pivot back towards the Pacific, and with Australia having an existing (and expanding) C-17 fleet, there may very well be some efficiencies that otherwise would not exist.
From my POV, the C-17 is an overall more capable air transport in areas NZ needs and likely wants. The questions I have, is whether or not what the C-17 brings to the table beyond that of the A400M is sufficient and sufficiently valuable, to justify the extra upfront costs. That and how long the NZDF can safely wait to replace strategic airlift, before mission failure rates become unacceptable. If the NZDF has a safe cushion of several (5+) years before it would become an issue, then the A400M is IMO a more likely selection. However, if the window is less than 5 years, especially if it is significantly less, then the C-17 is IMO a much better option.