Safety Pilot on a practice instrument approach reported a NMAC with another aircraft in the busy traffic pattern of a non towered airport resulted in a missed approach to avoid further conflict.

Narrative

Flying back to ZZZ1 with a friend who needed an approach for practice. Talked to ZZZ Approach for the RNAVX to ZZZ1. Was cleared; established; and tracking inbound when ZZZ Approach said; 'Traffic at your 9:00; also looks inbound; Squawk VFR; Frequency change approved. I announced our position on Unicom at a 5 mile final for Runway X; which was in use. The other traffic was a flight of 3 also inbound to ZZZ1. They were a formation flight. Their leader asked me what our ground speed was and I told him. He said; 'That will work out fine.' I saw them off our left and it looked like they were going in front of us. The usual procedure for these formation flights is to do a break over the runway. I figured we would follow and keep descending. I figured they would stay at 1;000 ft. and break just as we were touching down.The flight leader knew we were on the RNAV approach and said this would work so I believed him and we continued. I called a 2 mile final and thought I had them in sight. I heard someone call base at about the same time I heard a traffic alert on the ADS-B. I told my friend that I had the airplane and I did a go-around. While on the go-around I had another near miss. I called crosswind and traffic also reported being on crosswind. This was the first time I had heard him. I looked up and he was about 200 ft. above us and slightly behind. I told him that I would make a close pattern in front of him and he followed. This is an uncontrolled airport that has more operations than some towered fields. Next time a formation flight comes in I am clearing the pattern and waiting until they are all on the ground."