A TRACON Controller reported they issued an aircraft conducting an instrument approach in IMC a Low Altitude Alert when they descended below the glidepath but the aircraft had switched to the Tower frequency on their own.

Narrative

At XA:12Z I cleared Aircraft X for the ILS approach to Runway XXR via heading 150 and to maintain 3;000 feet until established. They read back the approach clearance correctly; even though it sounded like they used the callsign Aircraft Y. There were no other [company] aircraft in the airspace. I then told Aircraft X to maintain his current speed which was 190 knots; which he acknowledged. Aircraft X reported established on the approach at XB:14Z and I issued him a speed of 170 knots or greater until ZZZZZ. Since everything was issued correctly and read back correctly I moved on to my other tasks but monitored his speed every now and then. At XC:16Z I noticed Aircraft X speed dropping off drastically so I reached out to him but got no answer. I had plenty of other things to do so for a moment I moved on.At XC:16Z I issued Aircraft X a low altitude alert and to verify he was descending via the glideslope. There was no answer from Aircraft X at all and after a couple of transmissions I reached out to the tower to see if Aircraft X changed over to the tower without approval. He had switched over to the tower on his own and the tower said they also gave him the low altitude alert. I asked the tower to be sure he was going to be good on the approach since he was well below glide path and told the tower to climb to meet glide path or break off the approach. I didn't see Aircraft X climbing nor did I get a call back from the tower so I called them back at XD:17Z to make sure he was safe and going to be alright. The tower said Aircraft X said he was visual with the runway and the he was going to be A-okay. At the time this occurred the field was IMC/IFR and below basic VFR minimums. Unacceptable actions by the Aircraft X flight crew even after acknowledging everything correctly.Unacceptable actions by the Aircraft X flight crew even after acknowledging everything correctly. They didn't fly the approach correctly; they slowed well below the assigned speed and then proceeded visually to the runway when the field was below basic VFR minimums. They put the entire crew and passengers in danger while flying over downtown where the tallest obstacle building comes in at 967 feet and he was observed at times at 1;100 feet. I wasn't able to report this to management because there wasn't a manager around at that time of night. I doubt the tower gave him a brasher warning for possible pilot deviation.