Tolkien used a variety of different roots for “otherness” and “or” throughout his life. The earliest of these was ᴱ√VARA from the Qenya Lexicon of the 1910s, unglossed but with derivatives like ᴱQ. vára “other”, ᴱQ. var “or”, ᴱQ. varya “different” and ᴱQ. varimo “foreigner” (QL/100). Similar words in the Gnomish lexicon such as G. far(o)n “separate, different, strange” and G. faronwed “foreign” seem to be based on a distinct but possibly related root, apparently being derived from G. far- “separate, sever, divide” (GL/34). The Gnomish words for “otherness” seem to be based on the (hypothetical) root ✱ᴱ√ELE, such as G. el “or” and G. eleg “other, else” (GL/32); see the entry on ✱ᴱ√ELE for further discussion.
In the Early Qenya Grammar, the “other” words were based on ᴱQ. etya (comparative) and ᴱQ. nyanya (general), but these words were on a page of demonstratives and their primitive basis isn’t clear (PE14/55). The next published “or” word was S. egor from the King’s Letter in the omitted epilogue to The Lord of the Rings, written towards the very end of the 1940s (SD/129).
The next set of “or/other” words do not appear until the 1960s. The primitive form ✶khē̆ appears in notes on reflexives from 1965 as the basis for Q. hé “him, the other” (VT49/15). In rough notes on numbers written in the late 1960s, Tolkien gave the possibly-related root √KES “other”, with derivatives Q. exa “other” and Q. exe “the other”, apparently adjective and noun (VT47/40). Finally in some notes written in 1968 or later, Tolkien gave the primitive element √KHY- “other”, with derivatives Q. hye “other person”, Q. hya “other thing”, and Q. hyana “other [adjective]” (VT49/14).
These primitive forms also seem to be connected to various words Tolkien considered for “or” in the Ambidexters Sentence composed in 1969: khe >> hela >> hya (VT49/14). Patrick Wynne suggested the first two of these might be connected to 1965 ✶khē̆, and the last one to 1968+ √KHY-. This last root may also be connected to Q. ahya- “change” (circa 1960); if so Tolkien may have been vacillating among various possible forms throughout the 1960s.
Neo-Eldarin: For purposes of Neo-Eldarin, I think it is better to use √KHY- and its derivatives, since they are a more comprehensive paradigm including the best available Quenya word for “or”.
This root appeared in 1964 etymological notes as KEÞ [KETH] glossed “enquire of, question, examine something”, with derivatives like Q. ces- “to search, examine (something)”, Q. cesta- “to seek, search for”, and Q. cesya- “to cause interest, (lit.) to cause one to enquire”; the root was initially given as (rejected) KES (PE17/156). Given the root meaning “question”, it is probably related to the earlier verb ᴹQ. kesta- “ask” appearing in Quendian & Common Eldarin Verbal Structure (PE22/97), Quenya Verbal System (PE22/118, 120-122) and Common Eldarin: Verb Structure (PE22/138-139) from the late 1940s and early 1950s.