The NZ app will show any NZ flight regardless of plating carrier or booking source. The best way to have your flights show with third party bookings is to ensure that your AP# is associated to the NZ sector(s) correctly.

Assuming your AP# is correctly entered against at least one applicable sector, any NZ-operated/OAL-marketed (e.g. CX7402 AKL-HKG) and NZ-marketed/OAL-operated (e.g. NZ9601 LAX-LAS) flight should display correctly in the app as if they were any ordinary NZ flight (e.g. NZ001 JFK-AKL). It should also display OAL-marketed/operated flights (e.g. EK449 AKL-DXB) in certain circumstances and should be the case for both 086 and non-086 tickets. As akl_cat points out, the app (actually Carina as the app just displays what the PNR shows) does have an issue displaying OAL-marketed/operated flights. However, this issue will depend on your booking source rather than ticket stock. Ticket stock has little relevance to itinerary display in the app - it's more relevant to self-service functionality in the app (flight changes, ancillary purchases etc.).

The app will display all OAL sectors that connect from an NZ sector, provided they are without stopover. It will also only display a single OAL sector that connects immediately to an NZ sector. From recollection, the display time limit is either 12 or 24 hours between the OAL/NZ sectors. The reason is that your PNR, depending on booking source, will contain 'booked' and 'info' sectors. It's these info sectors that cause the issue. Info sectors apply to OAL marketed/operated flights in the NZ PNR and are added at the moment of PNR creation. Their sole purpose is to advise NZ of in/on-carriage for customer journey management. This also means that if you make a booking through a TA, you can potentially get separate tickets through-checked by NZ if they structure your PNR correctly......

In terms of your PNR booking source, there are basically two - direct and third party. Direct is 086 stock issued directly through an NZ booking channel such as their contact centre, website, ticketing office (more applicable to offshore markets), airport ticketing counter, etc. Third party is everyone else such as your local TA, online TA, OAL cash and reward bookings, TMC, cruise lines, Amex Travel, etc. regardless of ticket stock. Direct bookings should display all OAL flights on your itinerary as these flights have been booked in Carina and NZ have control over them. It's fairly uncommon for direct bookings to have info sectors but it does happen (normally where IRROPS is affecting the OAL and they need to rebook the pax to connect with NZ). With third party bookings, any connecting OAL flights will show as info sectors in the PNR. This is because these sectors were not booked in Carina nor does NZ have any control over them.

To try and demonstrate the above I created the below itinerary in Sabre (I've highlighted NZ for clarity). If this was a direct booking through NZ then all 6 sectors will display in the app. However, if this was a third party booking then only sectors 2/3/4/5 will. No info sectors for CX735 or BA58 will appear in the NZ PNR. CX735, despite being on the same through-journey as BA15 and NZ110, won't show on the app as it's not the carrier connecting immediately to NZ. BA58 won't show either as it's not part of the through-journey from NZ110.

1 CX 735Y 01JUN HKGSIN SS1 1430 1825
2 BA 15Y 01JUN SINSYD SS1 1925 0510 02JUN
3 NZ 110Y 02JUN SYDAKL SS1 0845 1355
4 EK 449Y 02JUN AKLDXB SS1 2030 0535 03JUN
5 EK 770Y 03JUN DXBCPT SS1 0910 1650
6 BA 58Y 10JUN CPTLHR SS1 1840 0540 11JUN