Air Traffic Controller reported aircraft descended below assigned altitude resulting in aircraft flying towards terrain. ATC canceled the approach clearance; climbed the aircraft and the aircraft diverted to another airport due to the weather.

Narrative

SR22 inbound to ZZZ snowstorm moving in from the west. SR22 was being vectored around the weather for the GNAV approach to runway XX. SR22 checked on with an old ATIS as we were changing runways due to weather. I gave SR22 the new information and advised him to expect runway XX. He requested to be vectored for the visual approach under the weather. I asked SR22 if he was sure; gave the latest PIREP which had light rime from a BE9L. He said in the affirmative; and I then vectored him into the downwind for the visual. The weather beat him in and the visibility dropped below 3 SM. I then told SR22 to expect vectors onto the final approach course behind a military aircraft. I cleared SR22 onto the final approach course maintain 7000 ft until established. SR22 appeared established I switched him to the tower. He read back the switch; but never switched. He came back to me saying that he needs to be re-vectored. He was descending below the MVA cleared on the approach. The altitude to cross at the FAF is 7000 ft. I canceled his approach clearance and climbed him as the low altitude alert was going off. I issued the low altitude alert. He then was vectored out of the weather and diverted to another airport.