General aviation flight instructor reported a near miss with another aircraft while on final approach. The instructor took control from the student and maneuvered to avoid a collision.

Narrative

This near miss incident happened at HQZ in the traffic pattern. I was in Aircraft X with a student from our flight school. The pattern got very busy during the second half of our visit. There had to of been at least 4-6 in the pattern/airspace at the time. I looked it up on flight radar and the offender was Aircraft Y. My student and I were following all instructions from ATC and as we were finishing up our base turn and about to turn final we received a traffic alert from TCAS for incoming traffic that was high. ATC also chimed in to say something; though I can't remember what they said as I was focused on the traffic heading right towards us now. I got them in sight and they were only a few hundred feet above us when I did; heading straight towards us in their premature base turn. I took over the flight controls from the student and banked hard left and added power in order to avoid an imminent collision. We were likely only 300-400 feet away; if that. We could both very clearly see the belly of the other aircraft and it would have descended right on top of us if we didn't move. It happened very fast and was way too close for comfort. I'm not entirely sure if tower understood how close it was either; honestly. After I maneuvered left to avoid them and get a safe distance away; tower asked if we would be able to follow in behind Aircraft Y. I agreed and tower eventually just told us to do a go-around as Aircraft Y pilot was still on the runway as we were approaching short final. Tower told this pilot that he was told to follow in behind me. The other pilot argued that he heard different and that he was number 2 to land. Though I don't have access to the audio from that day; I believe that tower and myself were in the right and Aircraft Y pilot made a very dangerous mistake. I know Aircraft Y had ADSB-out but am not sure if they had ADSB-in. Regardless I'm shocked that they never saw us or made any attempt to avoid a collision. This could have easily & quickly turned into a fatal event without our TCAS; towers traffic warning and my taking the controls for evasive action. One recommendation to potentially help avoid a situation like this in the future would be to encourage the FAA to allocate funds for this tower to finally get a radar system. This could have been helpful for the tower as they may have been able to warn Aircraft Y pilot that they were turning base too soon and asked them to continue downwind before it was too late. This also would have been useful for Tower to have a better understanding of just how close our two aircraft were to colliding.