Flight Instructor with student pilot reported an NMAC while maneuvering to enter the traffic pattern at a non towered airport. The instructor took control of the aircraft and executed an evasive maneuver to avoid a collision.

Narrative

Training flight from ZZZ to 2W5 for pattern work in Aircraft X.2W5 Runways 20-02. Executed standard entry overflying field 500 ft. above pattern altitude (1500 ms) from east to west to enter left downwind to Runway 02 on 45 degree entry. While overflying noticed one low wing airplane waiting for release; Aircraft Y and Z waiting for departure and one Aircraft A landing. When making teardrop turn to establish 45 entry to left downwind heard Aircraft Y making call and taking off from Runway 02 - intentions were to stay in the left pattern. Just after Aircraft Y another Aircraft Z made call and departed - intentions were to depart east. My student and I observing and hearing Aircraft Y turning left crosswind to Runway 02 continued our teardrop turn and announced that we will follow that airplane into downwind. We had visual contact and maintained proper spacing. While making 45 entry and turning left downwind to Runway 02 got warning from Aircraft A (who was waiting for departure from Runway 02) that I may have an airplane below me. After realizing that there is C172 Aircraft Z dangerously close almost below me I took controls from my student and started immediate climb and 360 turn to increase spacing and observe traffic below us. I would contribute this occurrence to the other C172 turning west instead of east and not making calls in the traffic pattern - there wasn't any call stating that they turn cross wind; and my own assumption that the other C172 will turn east from upwind leg.I believe I should be more careful with assumptions and take my situational awareness to a greater level and do not rely on others intentions but rather on dry observed facts. I think also that this situation would have been prevented if the other C172 made clear calls saying: e.g. N... taking off Runway 02; departure from downwind/base to the east.