PA-28 pilot reported experiencing engine RPM issues during climb to begin an ILS approach. Reporter stated reducing the throttle stopped the engine RPM oscillations and they were able to divert to the best suitable airport. Reportedly; the spring clip that secures the cable to the pivot bolt had come loose which caused the propeller control cable to detach resulting in the engine RPM issues.

Narrative

During a Instrument Proficiency Check flight; we stopped at ZZZ for fuel. While climbing to altitude to begin an ILS approach at ZZZ1; the engine RPM suddenly began oscillating between a very low RPM and an overspeed every few seconds. My initial thought was that we had a fuel delivery problem; or had received contaminated fuel; and verified that the electric fuel pump was turned on and changed tanks. This did not change the behavior.I tasked the Pilot Not Flying (PNF) (a CFII) with finding a suitable airport to land at (I think I said; 'Find me an airport'); while I continued to address the problem. Using the GPS he determined that we were about half way between ZZZ and ZZZ2. I opted for ZZZ2 as it offered a longer and wider runway; as well as maintenance facilities. He then programmed the GPS to display a direct course to ZZZ2.Knowing that the propeller governor had been overhauled during the annual inspection the previous month; I suspected a failure of the governor. I attempted to reduce the RPM with the prop control but found this had no effect. I then reduced the throttle to about 12 in hg to avoid further overspeed. This stopped the oscillations in RPM; but did not provide sufficient power to maintain altitude. Slowly advancing the throttle did not result in a reoccurrence of the oscillation; and I was able to restore level flight. The engine stabilized at about 2520 RPM; and manipulating the prop control had no effect.The PNF tuned in the AWOS at ZZZ2 and determined they were using Runway XX. He also tuned a radio to the ZZZ2 Unicom/CTAF frequency. I announced we were inbound with a situation; and would be landing straight-in on Runway XX. Other aircraft acknowledged my announcement and cleared the approach path and runway. I managed our descent to minimize the likelihood of needing to add power and landed without incident. I taxied the aircraft to the airport apron and shut down the engine.After removing the upper engine cowl I found the propeller control cable had become detached from the propeller governor at the pivot mechanism. It appears the spring clip that secures the cable to the pivot bolt had come loose; which allowed the parts to separate. After the cable was reattached to the propeller governor by an on-site A&P; the propeller control worked properly.In retrospect; I think the division of labor worked to our advantage. By delegating the identification of nearby airports; programming the course of flight; and tuning radios to the PNF; I was free to focus on keeping the airplane flying; identify the suspected cause of the failure; and find a workaround.However; I note that we did not attempt to notify ATC of our situation. One of the radios had been tuned to monitor 121.5; and both of us are familiar with discrete frequencies in the area which would have put us in immediate contact with ZZZ Approach. Neither of us are hesitant to declare an emergency; and both have done so in the past. Had we not been able to maintain altitude and been forced to land off-airport; this might have delayed or prevented search and rescue from locating us. I can only plead that both of us were busy addressing the problem during the short time between onset of the problem and the successful landing (less than 10 minutes).