If the small black wheel (tire, ring) on the bobbin winder isn't turning at all, it might be full of gunk (dried oil) on the inside around the axle.
If I remember correctly, the FW uses the same size ring, so you can try that, too, but it sounds to me like the axle needs to be cleaned.
I would try dripping Liquid Wrench or Kroil or some other penetrating oil into the oil port, then trying to turn that "tire" by hand. If you can break the hardened gunk loose and then just let it sit overnight, the penetrating oil can get into the cracks to soften up the old oil and lubricate the axle so that it can turn freely, again.
The old stuff will partially work its way out of the oiling port so that you can wipe it away and drip in some fresh penetrating oil. Just keep working on it and I'll bet it works fine within just a few days.
You may have to disassemble the bobbin winder to clean the axle area, but I think the penetrating oil will work well enough to avoid that. Good luck and have fun with your new baby. :)