Air carrier pilot reported the aircraft rolled uncommanded during pushback after the tug driver disconnected but failed to advise the flight crew to set brakes.

Narrative

XA:31 brake release flight ABCE. Last communication from tug driver was clear to start engines". Shortly after we started hearing the sound of the tow bar being disconnected. First officer stated that we were starting to move forward slightly. I hovered my feet on the brakes as a precaution. Parking brake set when it was clear that they had disconnected and we were moving slightly. At no time was there a "push complete set parking brake" communication received from the tug operator. The only reason we knew we were done with the push was the tug was pulling away from the aircraft and the pin was being shown to us. No further incident occurred and there was no known physical contact with objects or personnel on the ground. I called a safety timeout and requested communication with tug driver be established from operations. Driver admitted he didn't do what he was supposed to and apologized. Being concerned that the tug operator didn't understand the required communication procedures and to hopefully prevent a repeat event in future we requested to speak with a supervisor to let them know what had happened on the spot while it was still fresh. Proceeded to taxi without incident. Temperatures were extreme for ZZZ and ramp heat index was exceeding 100° F. Knowing disorientation from heat exhaustion can occur perhaps this was a contributing factor to the incident. I believe that the tug operator understood the seriousness of the situation and it will not happen with them again in the future."