I booked HAM-LAX with a connection at LHR. HAM-LAX with BA, LHR-LAX with VS.

Booking contains three different booking references: One from the travel-agency, another for the BA leg and lastly one for the VS leg. So different PNR means seperate tickets, right?

However, all three booking ref are accepted by BA and VS on their website - no matter which of the 3 booking ref I type in. And they display the entire HAM-LAX flight.

So, does that mean my booking is "on one ticket", despite different PNRs?

I'm asking because BA-Hotline says they do not check-through baggage for other airlines at LHR anymore - not even with OW members.
Yet, I've read here, that as long as the flights are booked "on one ticket", check-through is being/can be done.

Thanks for sharing your knowledge and experiences.