If ATC detects the pilot to be off route or that safety is compromised, they shall call go-around. If you continue the approach anyway, you better have a fuel emergency or a fire…
The ultimate decision resides with the pilot in command who can initiate a missed approach or go-around at any time. When you are cleared for the approach, you’re also cleared for the missed approach.
There’s a big difference being the only pilot onboard and having a co-pilot who monitors the flight, and handles comms.
I remember a solo nav flight during flightschool.
I had a class mate onboard, and that was a great reassurance. We were crossing a fjord and was pushed down by the weather. We agreed that we’d turn back if we were pushed below 2000’. At 1800’ we went IMC. So much for that 2000’ self imposed limit. 
I was at the controls and told my mate to look out while I looked in. I initiated a 180° as per our very limited instrument training instructions. We came out of the weather again and set course for home. Problem was that the weather continued to descend. We couldn’t get in the shortest way, so we turned around and went for the northern VMC route. This changed the alertness of the ATC! He started to follow our path in detail, calling out altitude and bearing and what the highest obstacles were, relative to our path.
I remember approaching base, without visual contact with the lights. It was a NATO standard lighting system, that can’t be seen from the sides.
I asked for verification of lights on at 100% and ATC confirmed, adding that I’d get the lights in a few seconds. He was correct and I turned final and landed with a sigh…
Having a friend and fellow pilot student, by my side helped me keeping cool, calm and collected. I like To think that I had been able to deal with that situation alone, but I know my heart rate would’ve been significantly higher and my palms wet…