lattesurf is right. It can't be done with software. It's about as do-able as getting the player to make beer with a firmware update. Now that
would be cool.
Am and FM are totally different. Adding an AM chip and other required parts wouldn't be that hard for the manufacturer, but the antenna is the real problem.
The AM broadcast band is low in frequency. This means it has a long wavelength and needs a long antenna.
The FM uses the earphone wire as an antenna. For FM, a wire that is stretched about 37cm makes a good 1/8 wavelength antenna, but for AM, it would be more like 37 meters long! Short Wave radio would also be very long.
To get around this problem, portable radios use a ferrite core antenna like this

Even though this is a very small antenna for the AM broadcast band, it's still too big to put into a Nano clone. The black part is made of iron so it would increase the weight of the player by about 50%. The player would have to be thicker to fit it inside and would probably be made longer in length as well. The metal back on the player would probably have to be replaced with plastic because the metal would shield the antenna. AM reception with an antenna that is this small would not be the greatest either.
This is why you will never see an AM radio that is small like this one
Dimensions: 37 x 27 x 10 mm
It's just the laws of physics.
I did find some real small ones on Ebay that were listed as AM/FM, but they were really only FM so beware.