How can I get this light fixture to light up?
The pictures are great. To my eye it does appear you've wired the 3-way switches correctly; I think you're right that neutral is missing. You can confirm by measuring voltage (with the multimeter) between black and white at the light with the power on. Measure voltage from black to ground there as well. Do the same check at the switch #1 box: measure voltage hot-neutral and hot-ground. For extra points, connect the light fixture and then measure voltage between neutral and ground.

It seems likely you'll find ~120 V hot-ground at both locations and some floating voltage hot-neutral. If you have the light connected you'd probably also see ~120 V neutral-ground. All of these conditions point toward neutral being open somewhere along the way. You're probably going to have to chase this upstream of switch #1.

Content of other boxes

I count four cables in the bathroom switch box. It looks like the pair in the middle are the switched lines to the light and fan; the outer cables must be the supply coming in and going out.

The hallway switch box also appears to contain four cables. Since there are 3 switches and no /3 cables, this box must be the end of the line for the circuit -- there isn't a cable carrying hot and neutral on to anywhere else. The neutral bundle looks a little suspect - the stripped part of one conductor goes down much lower than the others do, which makes me think it may not have been pushed up into the wire nut as far as it should have been. It probably just slipped down while you had the wire nut removed, and anyway this does seem to be end-of-line.

Focus on getting neutral working at the switch #1 location first. It's probably easier to reach than the light is, and you also have an always-hot wire there. By measuring the unswitched hot to neutral you avoid the risk of accidentally collecting false data (ie measuring at the light while the switched hot is turned off).

Could there be any outlets in this circuit? Verify neutral continuity through those too.

Tracing conductors

You may have come to the point of tracing out conductors. There are many ways one could do this, most of them laborious, and with various levels of risk of shock or equipment damage.


Measure resistance. Do this with a de-energized circuit. Disconnect both ends of a conductor and measure the resistance of the conductor. Do the same with other conductor(s) in the same cable. This helps to confirm that a cable goes where you think it goes, and that all its conductors have continuity. If the meter leads don't reach you can extend with some wire, an extension cord, alligator clips, etc. Could also measure loop resistance, for example: short black to white at one end and measure resistance between them at the other end, then try shorting black to ground and do the same.
Use a cable tracer. In the world of telephone and Ethernet there's a tone generator and inductive pickup kit often used to trace wires. This can be used for tracing mains wires too. Even though the tone generator can withstand the 90 volt POTS ring voltage, this probably should be done with the circuit de-energized.
Use a test lamp. This is done with the circuit live; appropriate precautions must be taken. One would open up the connections in the switch boxes, identify which cable brings the hot (and neutral) and start adding back one cable at a time, using the test lamp to look for hot and neutral showing up at other boxes.