Well, this is what this forum's all about! That's exactly what happened, the cell was at such a low voltage it refused to charge at all. Possibly a design feature that may well be fixed in other models, I don't know).
I took it apaaaart, four T6 self tapping screws, and measured the battery voltage, 200mV, so nowhere near enough. I set the lab PSU to 3.7V and current limited 70mA, and poked the battery terminals on the PCB connector. A few seconds of charge and she'd sprung into life. Charging current is 51mA.
Underneath there's a single cell LiPo 180mAh, you need to take off two more T6 screws and two Phillps head to access it, but it's not necessary for this process unless you want to change the battery.