Non contact testers are notoriously unreliable especially with runs of unterminated wire (no bulb or connection to the power supply).

Check the breakers. Check any GFCI receptacles for trip. The lights in the laundry aren't supposed to be on the receptacle circuit in modern codes, but anything is possible in older construction.

You are on the right track. The power is passing thru the lamp filament to the neutral and causing the neutral to rise to 120 volts because it is open somewhere between this point and the source. Start checking boxes, check the neutral bundles in the switch boxes to see if a white wire was barely in the wire nut and finally slipped out, or broke from improper stripping, check for blown backstabs at receptacles on that circuit. As soon as that current can find it's way back everything will work again.

Is it a 3-way circuit?