I'm not sure about the general case, but here in Argentina one of our biggest steel factories had a very big eucalyptus forest just to make coal. From https://es.wikipedia.org/wiki/Aceros_Zapla_S.A
> "Aceros Zapla" (ex- "Altos Hornos Zapla") was a steel forest mining complex located in the department Palpalá province of Jujuy, Argentina. It was the first steel center of Argentina [...]
> It had 15,000 ha of a forest of 30 million eucalyptus trees to extract the necessary coal in the process, which was the village Forestry Center. [...]
I don't know for sure hence the question mark :)
What I know that Carbon 14 which is used for carbon dating is a cosmogenic isotope which is created when high energy cosmic rays hit carbon atoms in the atmosphere and in other places.
Nuclear tests release radiation which created much more carbon 14 than what was created with the natural process of absorbing cosmic rays.
I'm not sure what's the current source of carbon for steel, I don't think it's coal it was always too impure and contain allot of contaminants especially sulfur, and I that high end crucible and powdered metal steels are made with considerably more pure and precise sources of carbon, from wikipedia it seems that the source of carbon for modern steel is charcoal (which is affected by C14 imbalance) and coke (which i have no clue if it's affected by post nuclear blast radioisotopes or not) which is refined coal but this only references to modern steel (late 17th century onwards, and I'm not sure what is used today).
Funny, that's the same reason I used my question mark!
If you looked it up then you're already ahead of me. Sounds like non-coal carbon is common. Although it sounds like coal could be used as a source of low-radiation carbon if you wanted to make new low-radiation steel.