It is a BA thing, another case of BA yet again refusing to abide by their contractual obligations in an attempt to inconvenience pax who choose to fly a competitor on a route served by BA. There are too many instances of this to be a mistake; it is clearly policy, to ensure you buy a BA ticket next time (or never fly BA again). Presumably your QF ticket is flying QF metal, in which case you have no problem, QF is happy to interline with EI at LHR. If you are on a QF codeshare on BA metal, then they won't accept the baggage. In any case be sure to have the ticket number for the EI ticket with you when checking in at SIN.