I am by no means an expert, and this is just guessing/thinking out a loud! But:
"When a new cell is inserted into the watch the capacitor is initially un-charged and looks like a short circuit. This means that there is only the sense coil in series with the base of the transistor to ground. This constitutes a resistance of about 2.25 k Ohm and so an initial base current pulse of about 0.5 mA flows (1.35-0.3/2.25K). This is easily enough to drive the transistor into saturation and send a correspondingly high current pulse through the drive coils. If we assume that only about 0.1v is dropped across the transistor then we can calculate (knowing the resistance of the drive coils is about 16 k Ohm) that the initial drive current is some 63 uA or five times the normal impulse. This gives a sufficient 'kick' to the tuning fork to start the movement. The circuit then rapidly settles into its steady state and the voltage on the base of the transistor rises to an average value of just above the turn on voltage (ie slightly more positive than would be needed to turn on the transistor)."
Which might suggest a hum before run? BUT, it is a mechanical drive, so as soon as the fork starts vibrating to produce the hum, then surely it should start turning the mechanical movement... even if slowly. Unless the movement of the forks was enough to produce an audible hum (was the pitch noticably lower than when running on full battery?) but not enough distance to move the pawl finger in and out of the index wheel to engage the teeth correctly to rotate the wheel!? The only other thought is that the hands were somehow stuck briefly!!?
(sorry not massively helpful, but an interesting conundrum!)