Nuclear orange (or atomic orange or radioactive orange) refers to very bright and very intense oranges favored by attention-seeking purchasers of athletic shoes, Corvettes, cosmetics, and other products.
As others have noted, there is an association of glowing with radioactivity, but as a child of the 1980s, long after advent of the Atomic Age, I would have referred to such an orange as fluorescent orange or perhaps DayGlo orange. The resurgence of radiological adjectives since then I would attribute to publicity about the dangers of Fiesta dinnerware, a highly popular line of ceramic dishware produced by the Homer Laughlin China Co. from the 1930s to 1972. Its vivid oranges (sold as "Fiesta red") were made possible by glazes containing uranium oxide or depleted uranium.
As noted in a Baltimore Sun article from April 23, 1994:
The uranium-bearing orange variety was discontinued in 1943 when the government placed wartime restrictions on uranium. Production resumed in 1959 when the restrictions were lifted and continued until 1972. In 1976, the company reintroduced the Fiesta line, but the orange color could not be duplicated with the new lead- and uranium-free glazes. …
Other brand names that used uranium-based pigments included Caliente, Early California, Poppytrail, Stangl and Vistosa, according to the federal Food and Drug Administration.
(Besides uranium, glazes may have also used radioactive thorium or potassium, and radioactive materials were also used in the manufacture of Cloisonné jewelry and canary glass).
The products containing the radioactive dyes were discontinued in 1972, due to changing tastes (according to the manufacturer), but the general public's skittishness about (and general ignorance of) nuclear radiation will help to ensure that calling a color nuclear or atomic or radioactive will add to its appeal, not to mention make ordinary plates and cups into collector's items. The most dangerous metal in the glaze, after all, is the lead.