BIL Tower Controller reported the weather and low visibility caused an aircraft conducting a go-around to come into close proximity to a crane near the runway.

Narrative

A crane was set up at Billings Terminal. The NOTAM stated 236 AGL. A snow storm began and visibility / ceilings dropped to IFR and near approach minimums. Aircraft X was inbound for ILS to Runway 28R. We were concerned about the crane and checked missed approach instructions for [the] 28R ILS. I told the Radar Controller to amend Aircraft X missed approach instructions from 4;000 before a left turn; to 4;500 before a left turn.Aircraft X did go missed approach and began a climb straight ahead. I noticed on the radar display that they were drifting left of the runway towards the crane; which was barely visible from the Tower. I gave Aircraft X a vector away from the crane as I was concerned with the proximity. I then saw Aircraft X out the Tower window pass just slightly right of and slightly above the crane. Reviewing Falcon and estimating the crane position put Aircraft X 0.1 mile laterally and 450 ft. above the crane. There were no adjustments to approach minimums; decision heights; or minimum descent altitudes made aware to pilots or ATC.I have been made aware that construction like this and use of a crane in close proximity to a runway alters approach minimums and this should all be NOTAM'd. As far as I can tell none of this was done. The Front Line Manager and Air Traffic Management were made aware of the event and operations continued throughout the day with even worse weather. Although the crane was only about a third of a mile from the Tower and more than double the height of the Tower; it wasn't always visible. ATC was not informed of any changes to approach minimums nor were any alternate missed approach instructions brought up before this event. This should have been all planned and briefed beforehand and possible cancelled due to weather.