The manufacturers are careful to avoid specifying the minimum filament current on these battery tubes, though they will admit that 1.6 volts is the absolute maximum. However, a few of them show up in mil-spec versions where you can find usually a minimum filament voltage of 1.0 volts. Since the terminals are exposed on top, this is easily measured.
The simplest rule is: When either D-cell drops to 1.00 volts with the preamp on, replace all the batteries.
The circuit adapts itself to falling high voltage; I designed it to operate down to 24 volts but of course it may sound OK below that. My recommendation would be to replace the HV when it drops to 24 volts, which should normally happen about the same time as the D cells drop to 1.0. This matching of the lifetime depends a lot on the manufacturing of the particular batteries, and their storage history - and you have no control over either one. If I was using a Quickie a lot, I would probably get my supplies in bulk (Costco or Radio Shack or some place that has a high turnover rate) and calibrate the first couple sets.
You may be able to find an old-fashioned analog voltmeter and install it across the filament of one of the tubes. It won't deplete the battery noticeably. I went to Mouser, looked up analog panel meters, and the cheapest ones were around $10-$15 for 0.5mA full scale. Add a series resistor to get 1.5v full scale, and calibrate it against a decent digital meter to find the 1.0v point, and you're golden. Heck, at that price get two!