After a bit more review, I see that Lawrage's question is actually more complicated than implied at first. This is an unusual situation since EK doesn't normally partner with other large airlines like BA; the fact that Qantas and EK have an agreement complicates things and allows flights from these two airlines to appear on one PNR when that is unusual otherwise. Basically, interlining of luggage is not transitive - Qantas interlines with BA, and also with EK, which allows the ticket to be sold. But it doesn't mean that BA and EK interline with each other and are required to transfer the luggage without e.g. Qantas as an intermediary.
It seems the info in ExpertFlyer can't necessarily be relied upon for this, noting there is a distinction between the mere existence of an interline agreement for ticketing (which is what EF data refers to), vs baggage interlining, which is dependent on airline policy. Here's a good thread from FlyerTalk in 2019 about this exact issue of bags not being interlined from BA to EK when sold on third-party (i.e., QF) tickets that otherwise might imply it could be done:
https://www.flyertalk.com/forum/emi...6-ek-ba-through-check-separate-tickets-3.html (See post #10 onwards from Isochronous for the part most relevant here, because it's talking about the situation where all flights are on one PNR.)
I realise it's three years old, but other threads seem to indicate that BA and EK are still at odds and still don't have any agreement. I would recommend calling BA as the first operating carrier and asking if they will check through, hopefully the phone agent will have access to the correct information. Lawrage, how much time do you have in London?
Note: Assuming this same issue would exist in Frankfurt, that London to Dubai itinerary proposed with only 1hr15 to collect luggage and re-check, doesn't seem tenable. I wonder how many people have gotten caught out by this kind of thing. Another peril of Qantas classic rewards...