The hot furnace gases that are used for heat-treating steel oxidize the alloying elements, such as chromium, to form a thin surface film.
These surface films interfere with visible light waves to produce the colors that your correspondent mentions.
The thickness of the films determines the apparent color of the steel as it interacts with light of different wavelengths.
Thinner films, which are formed at lower temperatures, seem yellow or gold.
Thicker films make the steel appear light blue. The thickest films seem midnight blue and finally black.
Temper colors on clean, bare steel are actually quite fragile, and are quickly lost if rusting thickens the surface film by depositing layers of hydrated iron oxides.
Many parts of the hundred-year-old clocks mentioned in the question owe the durability of their temper colors to the practice of dipping the tempered steel in sperm whale oil. The sperm oil gives a transparent, waxy protective covering to the oxide films, preserving their colors for posterity.
Widespread use of this technique has had the obvious disadvantage of producing a serious shortage of sperm whales.