The crew of VH-KEQ contacted Melbourne Centre at their descent point, 30 NM south of Wagga Wagga and were advised that there was no IFR traffic. The crew completed their pre-descent checklist, briefed for a runway 05 VOR/DME approach and commenced to descend. At 28 NM Wagga and descending through 8,000 ft, the crew broadcast their position on the Wagga MBZ frequency. They received a response from the Wagga Aerodrome Frequency Response Unit, that indicated the frequency had not been used within the preceding five minutes. The crew of KEQ continued their descent and contacted Melbourne Centre at 18 NM and advised they were transferring to the Wagga MBZ. The crew joined the 05 VOR/DME approach on the 10 DME arc and reported their intentions to Melbourne Centre, nominating a time for an operations normal call. Following this transmission, the crew of a second aircraft VH-EKH, broadcast their position 7 NM south west of Wagga, approaching 3,000 ft. They reported having conducted an overshoot from the runway 23 VOR/DME approach and requested that KEQ climb to 4,000 ft to facilitate separation. Both aircraft subsequently landed safely from the runway 05 VOR/DME approach. At the time of the occurrence, the ATS Sector 2 and 8 positions were combined. The two crews' initial estimates for Wagga Wagga differed by 8 minutes and the controller's expectation was that the first SAAB would have already landed by the time the second SAAB transferred to the MBZ. The controller is reported to have recognised the conflict between the aircraft and had passed traffic information to the crew of EKH as they initiated a missed approach from the runway 23 VOR/DME procedure. Communication between the two crews was subsequently established on the area VHF frequency with EKH at 7 DME on the overshoot from runway 23 and KEQ on the 10 DME arc and inbound for runway 05. The controller did not pass traffic information to the crew of KEQ. Following this occurrence, ATS team leaders briefed controllers on the importance of anticipating aircraft separation when issuing traffic information.