An instructor pilot reported that his instrument student pilot had difficulty controlling the aircraft on an ILS approach but he did not take control of the aircraft after ATC issued low altitude alerts.

Narrative

This flight departed in VFR conditions. It was an instrument training flight. The student pilot was initially in control of the aircraft. Navigation mode was GPS to Runway 28 at ROC. When we were cleared for the approach weather was VFR. Shortly thereafter; it went IFR and an IFR flight was filed in the air and we were cleared to the ROC airport and I requested ILS Runway 28 instead of GPS Runway 28. The instrument trainee was still in control of the aircraft. As we were being vectored conditions went to complete IFR. The trainee was having a difficult time with both heading and altitude. I was directing him but he was slow to respond. He was going below the assigned altitude on final. I finally took control of the aircraft. We were given altitude alerts by ATC. At some point we were cleared to land. The runway threshold came in sight and we landed. The low altitude issue would have been avoided had I; as the instructor; taken control of the aircraft as soon as we entered IFR conditions. This was not a deliberate action on my part but under the circumstances I acted too late. The flight was becoming uncontrollable. The Tower requested I call them upon shutdown. The lesson is as an instructor; take control immediately.