Flight Instructor reported a near mid-air collision while in the traffic pattern and performed in a go around which resulted in a near mid-air collision with another aircraft.

Narrative

I was flying with an instrument student; doing the RNAV XX into ZZZ. There were two aircraft in the pattern; making left traffic for [Runway] XX. Both were on downwind. Shortly after we passed ZZZZZ on the approach Tower called to let us know there was Aircraft Y on the left base and cleared us #2 behind Aircraft Y. I had Aircraft Y on my ADS-B In receiver; but not in sight. The ADS-B track showed Aircraft Y on downwind when Tower called. It took about 10 more seconds before Aircraft Y turned base. At this point Aircraft Y was 400 ft. below and on a collision course with us on final. I told my student to cancel the approach and go missed. After confirming with me that I wanted to go missed his first turn was to the left; directing into the traffic pattern. He turned this way to head back to ZZZ1 where we had departed from. We passed over Aircraft Y turning final and starting to climb above the pattern. This also put us close to another airplane on downwind. Visually it looked like we cleared both aircraft in the pattern by 100-200 ft. As I was ensuring we had visual separation the closest I saw on my ADS-B In between trying to visually locate the traffic was the second airplane 200 ft. below us.While we avoided touching other airplanes that day; I cannot let us get that close again. In the future I'll make sure we have more space when we go missed; turn out to avoid traffic in the pattern; and think further ahead of the airplane to anticipate conflicts. Another factor is to better brief roles and responsibilities before launching into the air. Since I was handling the radios my student was not paying attention to where other airplanes were and expected me to keep us away from traffic. I should also be more assertive to keep us away from other airplanes and take control when the student puts us in a dangerous situation.