Just out of interest, could it ever happen that while the said controller was giving departure instructions on the Tower frequency, a May Day call was being transmitted on the Radar frequency but not heard as the controller was busy on another channel?
The modern VCCS (Voice Communication Control Systems) that most ATC units use allow the controllers to "couple" two or more frequencies together. This means they then act as if they were the same frequency. So if Radar and Tower were coupled, an aircraft on the radar frequency would hear all tower transmissions and vice versa.
It could happen that a controller was giving a departure clearance whilst an aircraft on the radar frequency was wanting to make a MAYDAY, but the pilot would hear the tower transmissions and have to wait for an appropriate gap to transmit his MAYDAY. No different to if someone on the radar frequency needed to declare a MAYDAY whilst a controller was giving turn/descent instructions to any other aircraft.