A Tower Flight Data Controller reported a NMAC when the Local Controller instructed an aircraft to turn base from a wide downwind in front of another aircraft behind them also on downwind.

Narrative

I was working Flight Data (FD)/Clearance Delivery (CD) and was Controller in Charge (CIC). Local had two aircraft in the left closed traffic pattern of Runway 25 and an airliner on final for Runway 04.Local Controller extended downwind of Aircraft X; first in the pattern with Aircraft Y following them. Aircraft X readback the extension and started to widen to the right. Myself and the Ground Controller pointed this out to Local. Aircraft Y did not widen out with them.After the necessary spacing with the Runway 4 arrival; local turned the base of Aircraft X. Aircraft Y was still in a normal traffic pattern. Local verified they had Aircraft X still in sight. Aircraft X passed over Aircraft Y with roughly 100ft of separation. Aircraft X did not get a traffic call because they were in front.Both myself and the Ground Controller instructed Local to turn either aircraft or do something to ensure separation. We were ignored and told 'Aircraft Y has them in sight.' This was an unsafe operation.The traffic pattern should have been adjusted as soon as Aircraft X was seen bending out. Local Controller should have listened to myself as the CIC or the Ground Controller about ensuring separation regardless of Aircraft Y saying they had Aircraft X in sight.