For some very unusual applications (like research, ion chambers, X-ray and test facilities) the background radiation from the air borne 1950-1960's test blasts was enough to cause instrument and calibration problems. (So, Russian and Chinese and US and UK and French "dirt" contamination was lifted into the atmosphere, spread around, landed in iron and steel, got recycled and remelted, and is still inside modern steel. Which gets remelted and recycled into more steel.)
Doesn't cause hot "radiation" problems to any real-world users, but it was too much for the sensitivity of what these guys were trying to measure. So they cut up WWII battleship armor plate, melted it separately from ANY other metals (because it had been originally mined and smelted before any fallout landed anywhere) and went off on their merry way.
US Navy and civilian nuclear plant alarms regularly went off from released radiation from Russian and Chinese blasts, so fallout is definitely detectable. Meaningful? Well, radiation is hazardous in too great a quantity, and you CANNOT detect it without special instruments specifically looking for radiation. It's better to have a specification that will (realistically) never be exceeded, but check it anyway; rather than not check it at all and just assume there is no radiation present. A recycled nuclear plant primary pipe WILL contaminate future steels.