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 first version of Quenya Personal Pronouns (QPP1) from the late 1940s had ᴹ√ETHE “other” as the basis for the “other person” pronoun ᴹQ. the, along with a rarer “[yet another] person” pronoun he (PE23/91), but there are no signs of the past 1948. The next published “or” word was S. egor from the King’s Letter in the omitted epilogue to The Lord of the Rings, written in 1948-1951 (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), probably a holdover from the 1948 pronoun he mentioned above (PE23/91). 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”.
Tolkien used this root as the basis for Elvish “to be” verbs for much of his life. ᴱ√NĀ “be, exist” first appeared in the Qenya Lexicon of the 1910s with derivatives like the verb ᴱQ. ná- “to be” and ᴱQ. nasta “existence, being, creature” (QL/64). It also had derivatives in the contemporaneous Gnomish Lexicon such as G. na- “to be” and G. nast “being” (GL/58-59). It appeared again in The Etymologies of the 1930s as ᴹ√NĀ “be” (Ety/NĀ²), along with an invertible variant ᴹ√ANA² “be, exist” with the derivative ᴹQ. anwa “true, real, actual” (Ety/ANA²); this seems to be the only point where Tolkien considered this root to be invertible. The root √NĀ “be” is mentioned several times in Tolkien’s later writings as well (PE17/93; PE22/147; VT49/27-28).
The root √NĀ was often contrasted with other roots of similar meanings. For example:
> Stem of verb “exist” (have being in primary world of history) was √EŊE, distinct from √NA joining adjs./nouns/pronouns in statements (or wishes) asserting (or desiring) a thing to have a certain quality, or to be the same as another (PE22/147).
There were also periods when roots other than √NĀ were the basis for “to be” statement in Elvish. See the entry on √EÑ “exist” for a conceptual history of these alternate roots and their derivatives.