Well, scratch that. I placed a tape into the deck while it was disconnected. I can't get it to eject, but it does play, rewind, and fast forward. There is a fairly loud hum coming from the deck, but the functions seem to work fine. Gotta figure out what's wrong with the load/eject function next.
So, when you connected to the battery terminals and fired it up, the tape loaded played, RW and FF....but wouldn't eject?
If this is the case, you have either bad belts, broken gear or both.
even though the tape is stuck insde, remove the top cover of the unit again and get a better look at the worm gear. In your original post with pictures, the first picture you posted is a close up of area to look at. From the photo, the worm gear is to the right of the black wheel gear. It's what moves the black wheel gear. You can't miss it. It's just that the worm gear is perpendicular to the wheel gear and very small. From the photo, I can barely see some teeth.
Not sure what the HUM noise is about. Maybe it's the motor spinning, but with bad belts.
I think your worm gear may be in good shape, but you probably just have very old belts. These units have about 3 rubber drive belts inside, if memory serves me right.
Also, these Nak car stereo units are more than 20 years old from production. If yours was stored for a long period of time, belts become no good very fast when they just sit. The worst thing for vintage car stereo cassette players of any brand is to sit around for years without use. The belts dry out fast and stretch easily from sitting.
Look inside again, visually inspect the worm gear very good and let all of us know. If your visual inspection of the gear has all the teeth, you've probably just got bad belts.
Do a search on AudioKarma for a NakDoc and contact him... He repairs these units for alot of us and is very helpful. If all you needs is belts, your cost from him would be about $100 or so for all new belts.