Flight Instructor reported having a near mid-air collision with another aircraft that was cleared to takeoff while they were on final approach at the approach end of the runway. The other aircraft flew 100-200 feet directly below them.

Narrative

Aircraft was 1300 ft AGL (VCT pattern alt. 1100 AGL) 3NM prior to approach end of runway crew called initial and was cleared for the right overhead runway 36. 1 NM from runway crew heard tower alert a private jet that that they had crossed the hold short for 36 without clearance. Tower ask the jet if they were 'ready to go' and then cleared them for takeoff shortly after. Aircraft X continued to the upwind numbers started the right overhead. Upon reaching 30-45 degrees AOB (Angle of Bank); observer called out aircraft; a TCAS alert went off and Instructor in the right seat saw the jet about 100-200 ft directly below. Crew continued the right turn and climbed. Upon rolling out on downwind instructor took the controls and asked tower 'Did you see that?' and the only response was 'Cleared to Land.' IP (Instructor Pilot) elected to do a lap at pattern altitude to make sure the crew was safe to continue event. IP took the first T&G (Touch and Go) and then continued pattern work to complete the first student but chose to RTB for second student.