Update: I put the tilt vial + diode bridge in a metal box, and set three ball-bearings on the bottom so it could sit on just three well-defined points, and not wobble uncertainly.
Having done that, I'm prepared to believe all the visible variations in the output voltage are real tilt measurements, and not some kind of noise. There is a slowly rising background curve, in this case just under 1 mV per hour, likely from humidity changes affecting the wood frame of the house. On top of that, a sharp step change whenever I walk into the room and sit at the desk with my weight flexing the floor, which returns to the previous curve when I leave the room. There are also some much smaller wobbles, which might well correspond to activity elsewhere in the house.
It's interesting to make such a sensitive measurement with a simple all-passive circuit, no transistors or opamps (apart from the Keithley 196 DVM measuring the output, of course).
EDIT: Seems I could certainly clean up the layout, for example that long red wire going to the smaller copper electrode on the right will shift the output many 10s of mV, just with a small bend towards or away from the ground plane. Looks like we are measuring femto-farads of capacitance change here, so the wires to sense electrodes should be an absolute minimum length.