Flight instructor on final approach reported a NMAC. They transmitted to the aircraft converging with them to go-around because the converging aircraft was not replying or complying with ATC instructions.

Narrative

Entering left traffic for XXL from the west of the field we were advised that we were following traffic on final. Shortly after; I asked if we were cleared to turn base because I didn't have traffic to follow in sight. We were told 'yes.' I heard the Tower twice tell Aircraft Y that they were following us and were cleared touch and go number two. About 1/2 mile final; I saw Aircraft Y on a short approach and converging with us in his descent. At the same time; I heard the Tower verify with Aircraft Y that they were following us and they said 'no; where is he?' I was prepared to move out of the way to the right if they didn't send him around. With the parallel runways at ZZZ and split frequencies in use this was a risky move to consider.I heard the Tower tell him to go around but Aircraft Y pilot didn't hear it or read it back because he was transmitting at the same time. The Tower did not ensure Aircraft Y received the instruction to go around. At this point; Aircraft Y was getting fairly close and I said three times on frequency 'Aircraft Y; Go around!' After the 3rd time I saw him climbing and we landed normally. I consider having to give instructions on the towers behalf evasive action. It was clear the tower controller was not going to ensure the go around instruction. Every time I fly I hear the same small number of controller's voices working at both ZZZ and TRACON. These controllers appear to be on position for extended periods of time and you can hear the fatigue in their voices. This situation likely could have been prevented if the Tower was properly staffed and there were more people assisting the Tower controller's actively working traffic.