In one of the Conan Doyle stories, one of the characters is spinning wool by the most primitive method, using a distaff and spindle. He calls a piece of the yarn so spun a "wire".
I have seen a lot of spinning, but never heard that word applied to yarn. But it stands to reason that the word "wire" was used long before there was metal wire, now the only definition given in the dictionaries.
AHD etymology, however, refers to Appendix, wei- :
Indo-European Roots

ENTRY: wei-
DEFINITION: To turn, twist; with derivatives referring to suppleness or binding. Also wei- (earlier *wei1-).
Derivatives include wire, vise, and iris.
I. Form *wei-. 1a. wire, from Old English wr, wire; b. garland, from Old French garlande, wreath, from Frankish *wiara, *weara, wire. Both a and b from Germanic suffixed form *w-ra-, *w-ra-. 2. Probably suffixed Germanic form *wai-ra-. seaware, from Old English wr, seaweed. 3. Suffixed zero-grade form *wi-ri-. ferrule, from Latin viriae, bracelets (of Celtic origin). 4. Suffixed form *wei-ti-. withy, from Old English wthig, willow, withy, from Germanic *wth-, willow. 5. Suffixed zero-grade form *wi-t-. withe, from Old English withthe, supple twig, from Germanic *withjn-.
II. Form *wei-, zero-grade *w- (< *wi-). 1. Suffixed form *w-ti-. vise; viticulture, from Latin vtis, vine. 2. Suffixed form *w-t- becoming *witt-. vitta, from Latin vitta, headband. 3. Suffixed form *w-men-. Mimbres, from Latin vmen, withy, wicker. 4. Probably suffixed form *w-ri-. iridaceous, irido-, iris, Iris; iridium, iritis, from Greek ris, rainbow, and ris, rainbow goddess. 5. Perhaps suffixed form *w-n-. inion; exine, inosine, inositol, inotropic, from Greek s, sinew. (Pokorny 1. ei- 1120.)

And since the first meaning of the root is "twist", that
fits perfectly a yarn made by twisting, or spinning. No metal involved!