An instructor pilot providing instruction reported they turned base in front of traffic Tower had told them to follow and encountered an NMAC.

Narrative

Instructor/Student in the Aircraft X were midfield downwind in the pattern to land at ZZZ when Tower called traffic as 'Aircraft Y ahead of you on final.' Instructor saw Aircraft Y turning base to final ahead but was distracted as the student was not flying a downwind pattern parallel to the Runway; but rather establishing an acute angle toward the final approach course. Instructor watched the Aircraft Y lining up on short final and told the student to turn base and expect a quick turn to final. Instructor and student were looking toward the airport when Tower called for us to make an 'immediate turn to the south' to avoid Aircraft Y on final. Instructor assumed control; cleared the area for a left turn and made a rapid; level turn away from the airport toward the south. During the turn we spotted a second Aircraft Y. It too was on final approach with us between it and the airport. We were about three hundred feet above the Aircraft Y and about 500 ft. away horizontally. The Aircraft Y passed below us as we continued our turn to rejoin the downwind farther away from the airport. Once established on downwind; Tower cleared us to land; not mentioning that the incident Aircraft Y had already landed and cleared the Runway. Tower admonished us for not reporting midfield downwind or waiting for clearance to turn base. Most of our experience during this Pilot's training was at non-Towered fields. The Instructor; too had been operating mostly at non-Towered fields and not at this busy GA airport. Lack of familiarity with this airport's procedures caused us to start our approach with inadequate communication with the Tower. This incident could have been avoided had the Instructor spent more time operating at a busy Towered field more often.