I had to go to work, so I left her to deal with it. She called the gas company, and they sent someone by who observed that the new regulator the last person installed was installed wrong, so a vent was pointing up and could collect rainwater, which could then freeze. And sure enough, it had rained and then frozen.
But the water heater still displayed a lack of enthusiasm, so we also changed the pair of D cells that power the control unit and ignitor and I cranked up the heat to max (I generally do this every year when the cold weather comes, but we hadn't had any real cold yet).
This morning, my shower felt nice and toasty and I intentionally dawdled to see if it would stay hot, and it did. So I'm hoping it's all well and good now.
Tomorrow, the contractor comes to replace our oil furnace...