A Flight Instructor conducting training at a non towered airport reported a NMAC with a succeeding departure turning crosswind at the same time as they were.

Narrative

I was in an aircraft with my student teaching her landings in the pattern. I was in a Piper Archer and there was another aircraft behind me in a tail dragger with the tail number Aircraft Y. On the ground; there were two occurrences where the pilot verbally was rushing me to take off when there was another aircraft doing a simulated engine out and another time where an aircraft that just landed was in the middle of the runway continuing to taxi off but was not yet off the runway.My concern is more about what he did or the lack of didn't do in the air. I took off and was extending my upwind for other traffic to enter and I verbally announced. As I called my crosswind; I also heard him starting to turn crosswind but could not locate him. I tried to make contact with him but did not answer. As we were making our turn to downwind; I noticed that an aircraft was getting very close and had to take immediate action (almost a chandelle type of maneuver) to avoid him. He states he never saw me at all. When I asked him where he was; he stated ground references but could not find me or other aircraft in the pattern. This is a concern for me as I believe he lacks situational awareness. He stated that he never heard me because his radio was turned down and I came in very low even though other aircraft in the pattern did not state this concern.