Flight Instructor on initial climb from a non-towered airport reported a NMAC with an aircraft crossing their upwind leg.

Narrative

Departing RWY XX at ZZZ; our aircraft came within a dangerously close proximity to another aircraft; Aircraft Y. We had just departed ZZZ RWY XX and we were climbing in the upwind about to turn left crosswind to remain in the pattern for practice. Upon initiating the turn at 1;600 ft. we saw another aircraft crossing through the upwind/crosswind at a 45 degree angle at approximately 1;300 feet. This aircraft had made no prior radio calls about approaching the traffic pattern or entering it. It was not until we reported our turn crosswind that this aircraft made it's first position report; which they reported as downwind; although based on trajectory it was 45 degrees off crosswind just below pattern altitude. We evasively delayed our crosswind turn; then proceeded to make a wide right 360 turn to create space in the downwind for this traffic; which was proceeding in the downwind at a 45 degree angle away from the field. The aircraft proceeded to do a wide base to final single turn to land at ZZZ. We proceeded to land following this traffic and departed the area following a full stop with taxi back to the active. The closest proximity was at the upwind to crosswind stage departing runway XX. The aircraft passed just below us through the upwind at about 1;300 to 1;400 feet. We were at a pattern altitude of 1;600 feet. The aircraft was close enough to see the tail number and close enough to see the pilot inside.