Tower Controller reported a NMAC.

Narrative

Aircraft X departed Runway XXR on a ZZZZZ departure which turns to a fix about 4 four miles west. At the same time; Aircraft Y was maneuvering in the northwest quadrant of the airspace adjacent to the field. As Aircraft X turned west and climbed; Aircraft Y began turning south into their flight path. They nearly collided. I was the local controller working Runway XXR. Earlier in session; the Tower TMC came to me and asked if I would be comfortable taking a point out from Approach Control for an aircraft that would be taking pictures over the approach end of Runway XXL/R and over the city boulevard area. She relayed to me that the aircraft would be staying north and west of the airport which meant that it would not affect our traffic except in the event of a go-around on Runway XXL/R. I asked what altitude they would be at; she responded with 4;500; and I said I would take the point out.As the aircraft performed its mission; I watched its path and altitude to ensure that it was staying within the confines that had been coordinated. Thus far; it had. At some point; the TMC handed me two printed pages of a sectional chart with arrows and numbers on them to depict the location of the areas to be photographed. All of the markings indicated that the areas would not be in the flight path of our departures except any departing on a heading of 290. When Aircraft X was at the runway; I cleared them for take-off. At that time; Aircraft Y was southbound along the west side of the field. As Aircraft X was on departure roll; I thought to myself that I may have to issue a heading to Aircraft X if Aircraft Y did not turn west as they had been doing throughout the morning as I had observed.I projected that once Aircraft X had rolled out westbound; Aircraft Y would be about one half to three quarters of a mile off their right wing. As Aircraft X was rolling/rotating; Approach called me and said that Aircraft Y would be beginning their southbound turn. I took this as initial coordination for an upcoming phase of maneuvers; not as notification of an abrupt turn that was seconds away. Beginning about [the] stadium; Aircraft Y turned to the west; and I was satisfied that the two aircraft would not be in unsafe proximity. As Aircraft X was upwind and I believe beginning their westbound turn; I issued traffic about Aircraft Y to Aircraft X. Up to this point; Aircraft Y had performed the same maneuver - beginning to turn out to the west at about [the] stadium - and getting what I considered to be uncomfortably close to the departures from Runway XXR; but our departures from Runway XXR and Runway YYL/R had not been stopped or restricted.I feel that when Approach called me to tell me that Aircraft Y was beginning their southbound turn; I should have informed them of the ZZZZZ departure I had rolling. Unfortunately; in the seconds after the call from Approach Control; I suspended disbelief that Aircraft Y would actually start an uncoordinated southbound turn that was contrary to what had been observed throughout the morning; what I was specifically briefed on; and what was depicted on the printed maps that were handed to me. Additionally; I heard no coordinating from the TMC or the local one Controller from the back of the room to stop departures for Runway XXR; as Aircraft Y's subsequent flight path would be directly crossing the departure corridor to the west - coordination that always involves stopping our departures.I am not aware of the coordination that occurs downstairs in the TRACON. The Approach Controller that called me about the southbound turn may have assumed that his TRACON TMC or supervisor had stopped our departures or coordinated in some way. That same Controller may have been briefed on one maneuver and Aircraft Y performed unexpectedly. Whatever the case; the aircraft was allowed to fly directly through our primary departure corridor at the last second in a manner that was completely unexpected by the Tower. I took the point out under the impression that the aircraft would not be in the departure's way. Again; I should have informed the Approach Controller about the ZZZZZ. That may have changed this outcome. Aircraft should not be allowed to operate below 5;000 ft. when operating over the field or in their departure corridors. Our go around procedure takes aircraft to 4;500 ft. at midfield; and then climbs to 7;000 ft. past that point. I believe there weren't any arrivals to Runway XXL; but it still seems like an unsafe practice. If aircraft are to be allowed in close proximity to runways; the Approach Controller should be absolutely sure there are not any departing aircraft before allowing another aircraft to enter a known departure path. I also think some assumptions were made by everyone involved that a complete coordination had been accomplished. As Aircraft Y began their westbound turn further to the south with each pass; someone needed to verify their intentions.