Adding to my doubt is the article's mention of "pre-1942 vessels." Why would that be the date? The first nuclear weapon was detonated in summer 1945, and I'd wager that contamination remained sufficiently low until sometime in the 50s. According to this page, there were only 7 nuclear test explosions in the 40s, for a total of 9 nuclear explosions all together:
https://www.armscontrol.org/factsheets/nucleartesttally
But even if a single bomb is too much, the date should be 1945.
Back at the time, there was not very much requirement for nuclear safety/contamination prevention... so if you want steel definitely free from non-natural radiation you'll need 1942 or earlier steel.
Supporting this idea, here's another article on salvaging Japanese wrecks which mentions low-background steel in the context of ships built in 1943 and 1944: https://www.theguardian.com/world/2017/feb/09/images-reveal-...