I just found the paypal with the order for extra items, sorry for missing that. I think I got confused because the paypal was sent in by a person with a different name, but yeah there is a note that its for you. I'll get those parts right out to you.
Regarding testing parts, we do not test every single electrical part. We test the servo motors and we flash the nanos (which in effect finds ones that are dead, about 1 out of every 50). We also configure the bluetooth modules, which finds ones that are dead on arrival (about 1 out of every 200). We visually inspect the servo controller as we insert the shunt on V+/Vin. About one out of every 200 of those has obvious soldering defects.
One thing you should check is whether you inserted both batteries in the correct direction. I know I occasionally will put one of the two going the wrong way, in which case the project gets 0 volts since they cancel each other out.
If you're getting no lights at all, like no lights on the nano, no lights on the bluetooth module, it almost certainly is a bad +5V or GND connection somewhere (assuming the switch is good and the batteries are inserted properly).
There is a small chance the on/off switch is bad but that's pretty rare. Does it feel 'clicky' when you turn it on and off? If it feels mushy, you feel no distinct "click", that might be due to overheating during soldering. We do check for that during bagging, but one could possibly slip through. i can send you a new battery/switch with your extra parts if it feels mushy.
If none of that seems wrong and you have a multimeter, seeing if you get +5v coming out of the BEC would be the first step I would take.
I guess I would also smell the bec. If it smells like burnt plastic, you've got a short somewhere (i.e. +5V and GND are directly connected). This could happen if you had something plugged in the wrong way and caused a short circuit, that will pull too many amps and overheat the bec.