General aviation pilot reported a near miss with another aircraft in the airport traffic pattern under ATC control in visual conditions. The pilot maneuvered away from the other aircraft to avoid the aircraft; then continued in the pattern and landed.

Narrative

I had been flying left seat with an additional private pilot. Their name is Person A out of the Nashua Airport in Nashua; NH. Person A and I had done some pattern work. I just switched seats with Person A and was flying in the left seat. As we were holding short of Runway 14 for departure; Aircraft Y was attempting to perform a power-off 180. Their attempt seemed unsuccessful in which they made a wide climbing turn over the airport and did a go-around. Shortly after; we were cleared for takeoff and Tower had told us 'no delay.' We complied and were only told to perform left closed traffic; which we repeated back to ATC. Neither pilot recalls him saying anything about following Aircraft Y or alerting us of its position. He could have; but we were expedited out with 'no delay' instructions. We were making a normal left pattern and turning downwind; we did not see any traffic; but during the turn started to notice our courses possibly converging with Aircraft Y. Aircraft Y; which was expected to be far in front of us and much further downwind since it is a faster aircraft; was above us by a few hundred feet and to the right of us. They were told to make a 360 to rejoin the pattern. They had made a wider pattern than usual; likely due to their go-around position. Neither me or Person A had this aircraft in sight and were not aware of it until they were closer. When I noticed them; I stopped my climb around 1000 feet and they made a climbing turn in the opposite direction. However; it appears that I cut them off in the pattern and didn't have them in sight until they were too close for comfort. We continued our pattern as normal and did another lap after this occurred. Neither Tower or Ground said anything after the 360 was made by Aircraft Y. However; the airport was very busy on this day and they were occupied. We spoke with the Aircraft Y pilots after flight. We said we had not seen them. Both parties admitted that it was partially both of us at fault for which we should have been looking out for each other's aircraft. They mentioned they were alerted by ADS-B in their plane. We agreed it was best if we personally talked to learn from the experience and see both sides of the story. This was a very friendly and understanding exchange since the airport was very busy. Overall; I think this occurrence was necessary to report for pilots to be aware of extra alertness in busy environments. Also; to learn that different pilots take different widths of pattern legs depending on their purpose of flight. Nothing was intentional and no collision occurred.