This root and its extensions √KWEN and √KWET were connected to Elvish words for “speech” for much of Tolkien’s life. The first clear manifestation of this root was as ᴱ√QETE in the Qenya Lexicon, unglossed but with derivatives like ᴱQ. qet- “speak, talk” and ᴱQ. qent “word” (QL/77). It also had derivatives like G. cweth “word” and G. cwed- “say, tell” in the contemporaneous Gnomish Lexicon (GL/28); the alteration of the latter to ᴱN. ped- “say” in the 1920s (PE13/132, 164) is one of the clearest examples of Tolkien’s introduction of the sound change whereby labialized velars became labials in (Early) Noldorin and later on in Sindarin.
In his earliest writings Tolkien also used ᴱQ. Qen (Qend-) plural Qendi (LT1/235, QL/92) and G. Cwenn plural Cwennin (GL/28) as the general name of Elves, as well as ᴱQ. Qenya as the name for their highest language. It is not clear that these words were directly connected to ᴱ√QETE “speak” in this earliest conceptual stage, however. In fact Tolkien’s use of the clearly unrelated ᴱN. Gwenn as an Early Noldorin word for “Elf” in the 1920s (PE13/146) hints that they were not connected.
The same was true in The Etymologies of the 1930s where Tolkien gave both ᴹ√KWET “say” and ᴹ√KWEN(ED) “Elf” without an explicit connection between the two (Ety/KWEN(ED), KWET). Both roots were also mentioned in the first and second versions of Tengwesta Qenderinwa from the 1930s (TQ1) and around 1950 (TQ2) as √KWET “say” (TQ1: PE18/50; TQ2: PE18/100) and √KWENED among roots for Elf-kindreds (TQ1: PE18/34; TQ2: PE18/84). The first clear connection between √KWENED and “speech” (as opposed to just the language of the Elves) was in the Outline of Phonology from the early 1950s where Tolkien said:
> It must therefore originally have been made direct from the simple base of √KWEN, of which the word ✱kwened, Q quend- is itself only a derived stem, and its original significance was thus “spoken, articulate” rather than “elvish”, though indeed at the time of its making the Quendi were the only people or creatures possessing articulate vocal speech (PE19/92).
After the point Tolkien regularly connected √KWEN(ED) to speech, but it is possible he came up with the idea in the 1930s or 40s and we simply don’t have a record of it.
The first clear mention of shorter √KWE as the basis for both √KWEN and √KWET was in the Quendi and Eldar (Q&E) essay of 1959-60 (WJ/392) and associated draft notes (PE17/138). In Q&E, the only survival of this most primal form was ✶ekwē > Q. equë meaning “quothe” and Q. eques “quotation, saying, dictum” (WJ/392). All other derivations were from √KWEN and √KWET, though (with the exception of the language name Quenya) the derivatives of √KWEN had more to do with Elves and persons than with speech explicitly. Tolkien’s commitment to this paradigm wasn’t entirely firm, since in OP2 he added a marginal note in green pen (which he used for ammendations to this document in 1970) that read:
> There is however in no Elvish tongue any √KWEN having reference to voice/speech and this seems to be a guess of the Loremasters, perhaps affected by √KWET “say” (PE19/93 note #114).
This note was struck through, however.