ATC Tower Controller reported a NMAC between aircraft on final and another aircraft in the base turn that was not on the Tower's frequency and talking to the radar facility. Aircraft in the base turn was sequenced back to the airport.

Narrative

I was on controller in command (CIC) position; training taking place on local control position. Aircraft X was on a RNAV XX VFR practice approach. Aircraft Y IFR was on a right downwind to Runway XX; 4 miles west of ZZZ airport still in communications with ZZZ [Center] at 030 feet. When Aircraft X was on a 5-mile final to Runway XX Aircraft Y turned right base cutting Aircraft X off. Aircraft X had traffic in sight but had a TCAS RA advising to hold altitude then was told to descend. Aircraft Y overflew Aircraft X by approximately 003-004 feet. Aircraft Y never established communication with the Tower. Per Letter of Agreement (LOA) with ZZZ [Center] aircraft should be transferred to ZZZ no closer than 8 NM from the airport unless late communication is coordinated. No coordination took place. When I called ZZZ [Center] regarding Aircraft Y they said they cancelled the approach clearance. Lack of communication transfer led to this event. Recommendation - Aircraft being vectored for a visual approach should be in communication with the Tower prior to starting a base turn. Proper enforcement of the LOA would have also resolved the issue.