TRACON Controller reported a departing cargo jet reported a NMAC with a VFR aircraft that was under the control of the Tower.

Narrative

My coworker expressed concern to me over a situation that happened only minutes prior; while I was out of the room. They described that they had released a Runway XX departure with ZZZ Tower; and that departure subsequently flew right at a VFR target (Aircraft Y) that the Tower was working. After discussion with a Controller at ZZZ after the incident; a new Tower Controller said the previous Tower Controller told the Aircraft Y to hold southwest of the airport. They believed the previous Controller's intent was probably to hold them southeast of the airport instead of southwest. The Aircraft Y followed the Controllers instruction; but it put them directly in the path of the now departing Aircraft X the Departure Controller's first words to the pilot of the departure aircraft was 'traffic alert'; and the pilot said they were responding to the threat. After reviewing the radar replay; the two aircraft appeared to only miss by 100 ft. vertically and .1 mile laterally. These values are the lowest that the radar replay will represent; so the targets could have been closer than what I observed on the replay. ZZZ Tower has daily cargo and passenger operations; many of which elect to depart or arrive opposite to the current runway in use. This opposite direction operation does add complexity; and in this case led to a Controller loss of situational awareness. Contributing to this loss of situational awareness was the lack of a Tower radar display in the Tower. I believe that if the Tower were to have an operating radar display; they would have been able to identify the position of the VFR aircraft; noticed their error; and prevented this NMAC.