I don't know if there's anything special that applies to dual citizens in these circumstances (relying on the expertise of other members of this site to add their answers if there is) but assuming you can enter either as a British citizen or as a New Zealand citizen and in each case you need that country's passport to prove citizenship, your options are:
No visa needed but you need to be a temporary visitor with a ticket for onward travel:
British passport holders can enter New Zealand as a visitor for up to 6 months on arrival without a visa, provided you can satisfy an Immigration Officer that you meet the requirements of the immigration rules. Visitors must have an onward ticket.
so that doesn't seem like an option here (because your visit isn't temporary and you don't have an onward ticket).
New Zealand Citizen
As you don't have a passport you'll need either an urgent passport or an emergency travel document:
If you don’t have a current passport and need to travel in a hurry you can apply for an urgent passport online.
Allow for up to 3 working days, plus delivery time. The passport is valid for 10 years for an adult and 5 years for a child.
Sometimes it’s better to apply for an urgent passport online than an emergency travel document. Often getting an urgent passport is quicker, depending on where you are in the world. It’s also less expensive, lasts much longer and accepted by more countries.
You could collect your urgent passport from London or get it delivered to you in France. For more info see https://www.passports.govt.nz/urgent-travel/
Bear in mind that the three working days will be affected by Christmas holidays:
The Passport Office will be closed from 5:00pm on Friday 23 December, and will reopen on the 4 January. During this time a Call Out service is available by appointment, please contact us for further information. The Contact Centre will be open on 28, 29 and 30 December from 8:30am to 5:00pm for urgent enquiries only.
The other alternative is an emergency travel document:
Emergency travel document when outside of New Zealand
If you’re overseas and don’t have a current passport and need to travel in a hurry you can apply for an emergency travel document. But it may be better to apply for an urgent passport online, as this is often quicker, depending on where you are in the world. It's also less expensive and lasts much longer.
Emergency travel document fee
An emergency travel document costs a flat fee of NZD$500.00 and you need to contact a New Zealand High Commission or embassy. You can’t get one in New Zealand.
Application form is here: https://www.passports.govt.nz/assets/Uploads/Forms/ETD-form.pdf
To get an emergency travel document you'll need to talk to - and possibly visit - your local embassy, which in France is at:
Ambassade de Nouvelle-Zélande, 103 rue de Grenelle
Tel: +33 (01) 45 01 43 43
Fax: +33 (01) 45 01 43 44
For more info talk to the NZ Identity and Passports Service on 00 64 (4) 463 9360. I'd suggest phoning them to explain the situation and see if they'll allow you exceptional permission to prove your NZ citizenship with a non-passport document - a British friend whose passport was destroyed in an accident was able to get back into the UK that way, but it did involve a whole lot of talking to Immigration before the flight and pre-warning the airport staff of his arrival so they knew what to do with him.
There's also a UK out-of-hours number open "between 4pm and 10pm weekdays or 8am to 10pm weekends and bank holidays" which is +44 20 7968 2730. Brace yourself for the £475 fee if you get a passport through them, it might be cheaper to change your flight to allow enough working days to get an urgent passport in the regular way. https://www.passports.govt.nz/passport-costs/