STEPHEN-
I have the same car, but wih a stick shift. So no neutral switch.
I don't see why you couldn't junp the neutral switch to verify that is (or isn't) the problem. If that doesn't identify the problem, I would search for bad connections. (I assume there is no shorted wire in the circuit, or you should have lost a fuse. and that the ignition is working if the car starts when you short accross the starter.) The '52 book on this website has a usable wiring diagram if you lack a manual. ('51s and '52s are identical except for trim superficialities.)
'51 Special
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