An Interline agreement covers journeys involving more than one airline on a single ticket. Travelling on two tickets is not an 'interline' journey.
So, you have a single ticket with the routing GVA BA LHR NZ AKL, BA at GVA will check the bag through to AKL. The contract you have is for transport of you and your checked baggage from Geneva to Auckland.
If you have two tickets GVA BA LHR; LHR NZ AKL this is not an interline journey. BA do not have to check the bag through. Their contract with you is to deliver you and your checked baggage to LHR. They could check it through to AKL, because all the systems are in place, but they chose not to.
If you have two tickets, check with the first airline before you arrive at the airport, so you will know what their policy is. Double check on your return because you likely will be checking your bag with a different airline. It could work one way but not the other.
Note - this is a commercial decision of the airline, not a technical one. It's almost always technically feasible, but if head office has decided they are not going to do it, they are not going to do it.