Loosely inspired by this closed rpg.stackexchange question titled "What is Cold Iron actually? — Forget what it is; let's talk about the origin of the set phrase "cold iron" in English!
Rudyard Kipling's poem "Cold Iron" (c. 1910) may well have popularized the exact phrase in today's pop culture, but it (1) is extremely recent and (2) isn't directly related to the fair folk.
This great rpg.net thread points to Robert Kirk's The Secret Commonwealth (written c. 1691 but published only(?) in 1893), chapter 1, which uses the exact phrase:
The Tramontains to this Day put Bread, the Bible, or a piece of Iron, in Womens Beds when travelling, to ſave them from being thus ſtollen; and they commonly report, that all uncouth, unknown Wights are terrifyed by nothing earthly ſo much as by cold Iron.
So, did Robert Kirk actually coin this poetic turn of phrase? Or can it be traced back farther?
The defining characteristic here is that we're not just talking about how supernatural beings dislike iron; we're talking specifically about textual sources that describe the elf-repellent iron as cold (regardless of what you think the source means by that).