Online check in for connecting flight (different airlines)

I'll be flying with Air New Zealand from LAX to LHR and then from LHR to PRG with British Airways. I have both bookings available under the same Air NZ booking reference and they are all visible in my Air NZ mobile app.

I'm not sure, however, how the online check-in works in this case. I spoke today with a lady from Air NZ customer service and she said that my baggage will be transported from LAX to PRG and that I won't need to reclaim and re-check my baggage at LHR, but that I will need to check in separately for the other flight via the app. How exactly does this work? My main concern is:

• When the second online check-in asks me about my checked baggage, what do I say? Since my only checked bag will already be checked in from LAX, do I just say that I don't have any? Any chance it will cause the airline to offload the bag?

Also, am I required to print out the boarding pass when checking in online? I believe I used a boarding pass available from an airline app with a QR code when I once flew with Ryanair.

Any info about this would really help me sleep easier. Thanks!