C152 Flight Instructor reported their student turning crosswind early resulted in a conflict with an aircraft they were to follow. ATC issued a turn to avoid the traffic; Flight Instructor took control and turned to avoid the traffic.

Narrative

I was doing pattern work with a student at ZZZ; we were making right traffic for Runway XXR. Approximately 1.2 hours into the flight; we became dangerously close with another aircraft in the pattern (Downwind). This was the traffic we were supposed to be following; however we turned crosswind too early causing a conflict. Tower told us to turn East immediately; with a clear sense of urgency in his voice. At the same time he said that; my eyes met the conflicting traffic and I took over the controls and evasively turned us East. I think the main issue is that I lost situational awareness. ZZZ is a demanding airport; and I should've been just as vigilant on the upwind to crosswind leg as the base to final. I'm a new instructor with less than 10 hours dual given and this is the first time I've taught a new student landings; I think my focus was diverted away from the crosswind leg and focused more on the base and final leg and touchdown and how to coach the student on this. I need to be cognizant of all phases of flight whether my student is flying or whether I am.