Here is a little information from the Chevrolet Service News Bulletin (January 1954) about the
neutral safety switch (pages 3 and 4). I will check out a wiring diagram tomorrow and I will also check out how my push button is wired. My guess is that the hot ignition-switch lead goes to one side of the dash push-button starter-switch. The other side of the push-button starter-switch goes to one side of that neutral safety switch on the shift column (in the engine compartment). The other side of that neutral safety switch on the shift column goes to the starter solenoid.
If you have no wires on the neutral safety switch, then I would guess that your push-button starter switch on the dash goes directly to the starter solenoid.
I am just now learning about these '54 Hydra-Matics. I drove mine tonight (for the third time) after tuning the carburetor (mostly correcting misconnected vacuum lines by the previous owner and bringing down the idle speed and setting the mixture screw for maximum vacuum and smooth acceleration)). It shifted up and down much more smothly after these corrections.