Well, all terminal in MEL are connected, from T1 (Domestic QF) to T2 (International) only need two-minute indoor walk. About customs in MEL, basically all pax (Except F/J/Frequent Flyers) need to use e-passport lane.
Both QF and EK will check your bag to LHR.
Bottom Line: Which flight you are traveling on? QF9 or EK407?