Here, for what it is worth, is what I know, and have been telling people for ages.
A separate ticket, on a separate booking is a separate contract: Bags cannot be checked through between tickets/contracts, As you can imagine, this issue arises often on flights from AKL to LHR.
The reason for this is that the airline that accepted the bags initially, is taking responsibility for both you, and the bags, all the way through to CFN. Were your bags to be lost or late on the last sector they would be liable, and would also be liable for accommodation costs in a delay, since responsibility had been accepted for the entire journey.
You may get lucky if the check-in agent makes a mistake, but I would be surprised. Please let us know how you get on.