Welcome to Flyer Talk chris535is

As welly points out, most of this is about through passengers flying between London and Auckland.

For you, and assuming you are an American, things are a bit different.

Going to AKL you will have a change of terminals at LAX (NZ flies out of T2, DL uses a different terminal) and need to reclear security (as there is no airside transfer). If you have *G status or are in business class or have paid UA Red Carpet Club membership, then you can use the very nice NZ lounge while you wait.

Coming back from AKL you first clear immigration. As American this is much quicker and easier than for non-Americans. Then pick up any checked baggage and take them through customs. If tagged through to ATL you can use the bag drop just after customs on the right hand side (or carry them over to DL's terminal if you prefer - such as tight connection or wanting to standby for another flight). If not tagged through to ATL then you have no choice but to carry them over to DL's check in. Change terminals, clear security and wait your onward flight.