EK won't let you in the lounge as access is only based on departing flight. EK flights depart from concourse A, B and C, whereas BA flights are only from D, but since you have such a long layover it shouldn't be a problem.
Unfortunately, you cannot buy access to the EK lounge as you are not travelling on EK, and you must be a skywards member too. The only other option you have left is to pay for access to the Marhaba lounge which is about AED 200 for 3 hours I believe. Though it can be heaving and not a pleasant place to be especially during your layover as it is rush hour in DXB.
Just to check, is this booked on one PNR? Because you might have to recheck your bags, I'm not familiar with QF's policy, though they should through check with EK and BA too due to their partnerships.