Air carrier Captain reported a critical ground conflict and missed approach when a non communicating aircraft landed on a non-active intersecting runway.

Narrative

While on a localizer approach to runway XX ATC had asked us if we could cancel as soon as possible as they had another aircraft right behind us on the approach. I told my FO I would cancel as soon as we broke out and had the runway made. We saw an aircraft on TCAS at 1;400' in what would have been an appropriate position for the left downwind on runway XX. As we broke out on the localizer approach I did not have the aircraft in sight. I made several position reports with miles out; and time at 35;15;10;5; and 2 Miles. We never heard the aircraft on TCAS make a single radio call. Once we had a visual on the runway I started looking for that traffic. It appeared again as if they were still on the downwind leg and their altitude was still showing 1400'; at that point we canceled IFR with center. As we crossed the threshold of runway XX I saw them crossing the threshhold of RW XY. I made another call hoping they would go around and finally a call was made that they were Rolling out on runway XY" while they were still in the air. We initiated an immediate go around. The center frequency was too busy to get a word in and at this point we were a VFR aircraft. The ASOS at the time was reporting 330 at 8; 10 SM; few1100'. I took the controls from my first office after the go around and flew the aircraft back around for a visual approach. We did get a "DON'T SINK" warning on the North West side of the airport. We did have all terrain and obstacles in sight. Our altitude was 1200-1300 AGL to remain clear of the clouds. We were also in communication and had visual contact with the traffic that was on the approach behind us. Cause: This event was caused by an another aircraft not making radio communication and landing on a runway that was not the active runway; also by canceling IFR with center after assuming the position of VFR traffic. Suggestion: I feel like this event could have been avoided by just waiting to cancel IFR when we are safely on the ground; and not be succumbed to the pressure ATC."