Estimating maximum current capacity of non copper wires
The limit for copper and nichrome wires is different in character.
Copper wires are designed to carry current without getting too hot. As such, they are rated assuming a temperature rise that will allow the insulation on them to remain intact. The rated tempertaures are usually sub 100 °C.
Nichrome wires are resistance wires, and are one of a series of alloys that are designed to get hot, and stay hot, without undue shifts in resistance and corrosion from gasses in the atmosphere. Depending on the specific alloy, it may be possible to have temperatures of the order of 1000 °C.
Thermal conductivity is not particularly relevant here. It's only when a short piece of wire is being used between large terminals, like a fuse link for instance, that the thermal conductivity limits the temperature rise of the centre of the link, due to the action of the nearby terminals as heatsinks. Both copper cables carrying current around, and long nichrome wires cutting plastics, are so long that the heatsinks at the end are irrelevant. The principal heat loss mechanism is convection and radiation from the surface of the wire.