Replacing the stock 2 conductor power cable on my Nakamichi 582 cassette deck?

If 2 wire it is most likely to have double insulation so no ground needed.
While this would not be a compliance safety ground, you could always try a three wire config to the metal part of the transport via a toggle switch and/or a ground lift resistor.

Should I just replace with a much heavier gauge of cable and leave in the 2 conductor non grounded as OEM supplied configuration.
or should I replace with a 3 conductor and add a ground and if so where would be best to add it at the cassette deck end? Or would that be a move that may actually induce some type of ground loop/hum?

I would not add the chassis ground. The power wiring for the Nak is class 2 double insulated. The piece of equipment was designed not to be grounded, connected to the mains power equipment ground. Grounding the chassis could end up doing more harm than good to the SQ of the unit.

Problem with installing a bigger diameter power cord is the inlet captive plastic grommet hole would need to be enlarged. Doing so will pretty much ruin any future vintage value of the Nak deck.

If you think you can improve the sound of the deck you might consider installing an IEC male connector on the existing power cord of the deck. That way you could try different aftermarket power cords. You would just cut the cord off about 6 inches from the back of the deck and then install the IEC connector.

I would suggest before you start, whatever you decide to do, is to check the power cord, (Actually checking the primary winding of the power transformer), for the proper AC polarity orientation. You will need a multimeter for the test. I assume the plug on the power cord for the Nak is not a polarized plug.

Look at the power cord for an identification of the identified conductor.
It might be a continuous raised ridge/s on its’ side the entire length of the cord.
It might be a white continuous strip on its’ entire length.
It might be lettering, printing, on one side of its’ entire length.

Orientate the non polarized plug so the plug plugs into the wall outlet so the identified conductor is on the neutral side of the wall receptacle outlet. (The taller slot hole contact of the receptacle if 15 amp. "T" slot if 20 amp.)

Make sure no interconnects are connected to the deck.

Power on the deck

Set your multimeter to auto volts.

Touch one meter probe to a bare metal part on the deck. You may need to remove a screw. Touch, insert, the other meter probe inside the "U" shaped equipment ground contact of the wall receptacle. Note the voltage.

Turn off the deck. Unplug the plug for the deck and rotate it 180 degrees and plug it back into the wall receptacle. Turn the deck back on. Measure the voltage from the deck’s metal chassis to the wall receptacle equipment ground again. Note the voltage.

The correct polarity orientation is the lower voltage reading of the two. If Nak did their job correctly and made sure they had the polarity orientation correct your first voltage reading will be the lowest reading.

Jim

@jea48 
I was sort of thinking if it was designed to work fine without the 3 wire connection, then adding a ground could be more harmfull sq wise which is part of the point of asking for more expert advice than my own. 
So with that being said I could at least investigate the possibility of adding larger gauge power cord, I would not be too worried about enlarging the hole in the back a little. If I use a rubber grommet it would look very similar to stock anyway and for what I paid for it I will most likely not part with it.
Or your idea of adding a male IEC connector could be done but if I cut the original cord down to approx 6 inch surely that would have a similar effect on potential resale value as well?
Not that I see that as any issue to myself tbh.
As far as polarity excellent point and one I have already verified but always a valid starting point.
Dont get me wrong the sq is pretty good but does not mean it might not be better still with a better power cord.

@uberwaltz 
Did you ever change the power cord? I'm wanting to do the same thing on may Nak CR4, I'm really curious to see if changing the cord makes a big difference.

No I did not go any further on this matter...yet...lol
Had plenty of other changes keeping me busy

Before reading this tread I wasn't sure if I should go with a grounded cable or not, but now I know to go with a non-grounded.

I think I'm going to try to put an AUDIOQUEST - NRG-1.5 C-7 on mine, it may take me a couple of months to get a round to it.  If I bet you to it, I'll let you know what happens.