Why would a 120v circuit work without a neutral?
It's using the earth as a return path

You really don't want to do that.  Dirt doesn't conduct electricity very well.

Your neutral is bonded to ground in your main panel.  The power company's transformer also bonds neutral to ground at the pole, and all your neighbors have neutral-ground bonds like yours.   The current is trying to return to source (neutral) via your ground rod, wet dirt between your homes or the transformer pole, their ground rods, and back to neutral. 

You're measuring it and saying your hot is varying 0-60 volts.  No.  Hot isn't damaged, so it's exactly where it belongs at 120V.   Neutral has come UP to 60-120 volts!  And ground has come with it!  Because neutral is using your neutral-ground bond in the main panel, and lifting ground.   Which means your neutrals and equipment grounding conductor is floating 60-120 V above actual proper earth.  That's a good way to electrocute somebody because it means the grounded bodies of appliances or grounded equipment are also floating up to 60-120V!

Shut this off right now until it can be fixed.