An Air Taxi pilot reported a NMAC with another aircraft which was not responding correctly to ATC instructions.

Narrative

While proceeding northwest on a VFR flight from ZZZZ to ZZZZ1 we began our descent into ZZZZ1. I was Pilot Monitoring in the right seat and my Captain was Pilot Flying in the left seat. While descending through roughly 3;000 to 5000 ft. Approach notified us of IFR traffic; 8-10 miles away departing ZZZZ1 that were proceeding southbound and climbing. We did not have the traffic in sight and advised that we were looking. My Captain and I identified the target on the map page of the GPS and could see that we were on a collision course but had plenty of time to take action so we continued our descent. Shortly after we stated that we did not have them in sight; Approach gave them a left turn to 270 from their initial southerly heading and repeated it to clarify that they needed a wide left turn due to traffic. We heard the other aircraft acknowledge the instructions correctly and believed that the initial collision hazard had been mitigated. While continuing to monitor the target on the map page of the GPS; my Captain and I noticed that the aircraft's course had not changed and we were now about 5 miles apart and on a collision course. The other aircraft was still climbing southbound and we were still descending north-west bound with somewhere around 1500 ft. of separation. My Captain suggested a turn about 20 degrees left to add separation for when they began their left turn and I agreed that that was a smart decision.While executing the left turn; Approach advised the other aircraft that they needed an immediate left turn to heading 270 for traffic. We heard no response from the other aircraft. At this moment we were about 2 to 3 miles apart; still on a collision course and likely closing in on each other somewhere around 350 kts. The GPS had begun to alert us about a traffic hazard and we still did not have visual contact with the target due to the scattered cloud layer ahead. With their lack of complicity; I suggested that we turn right immediately to the north/northeast to avoid crossing through their southerly course. My Captain agreed that that was a good idea and they disconnected the auto pilot and immediately began a turn to the right to avoid the other aircraft's southerly course. As we started to turn away from their southerly course the other aircraft began the left turn that they did not acknowledge or initially comply with. We had no way of knowing this because we couldn't locate the other aircraft visually and the traffic warning was not accurate enough to depict immediate turns from other aircraft. It wasn't until we were about a 45 degree bank to the right that my Captain told me that they saw the other aircraft pass underneath our left wing by roughly 500 ft. At no point in time did I ever see the other aircraft due to their position and visibility restrictions low and left from the right seat. Almost immediately after or before the aircraft passed underneath our wing; I don't remember the exact timing; but the other aircraft stated hesitantly that they thought they were on a 180 heading. Approach corrected them and reminded them that they said 'left turn heading 270' multiple times. Following the near miss; the flight continued uneventfully and the passengers did not seem concerned or aware of what happened while deplaning.Thinking back the best suggestion to have avoided this would've been to level off and stop our decent sooner or alter course sooner. However in the moment by the time realized they were not complying we were close and somewhere around 1500 ft. apart. Slowing down and picking the nose up could've been the difference between spotting them and having an aircraft climb into us right in our blind spot especially if they were climbing at or near their best rate. Altering course more aggressively earlier also would have helped but after hearing them correctly acknowledge instructions our initial deviation to the left felt like a smart choice with safe margins in the moment. Operationally; we tryto remain efficient and safe so based on what we first knew; turning very far to the left would have put us very far west and likely could have put us outside of glide so it was an impractical choice assuming they did as they acknowledged.