A sad +#StoryOfARoot that is, a story of forgetfulness and forgetfulness and oscillations. Curiously, the situation in the Lexicons is remarkably similar to what we are going to obtain decades later, with the alterations really going full circle.
* 1910s The Lexicons distinguish two separate roots: TEKE 'make marks' (QL:090, GL:040, 068-9), with such derivatives as 'mark, write' (teke, tectha), 'pen' (tenka, tenc), 'book' (tekka/tekte) and 'letter, mark, sign' (tet, taith); as opposed to TEÑE 'feel, understand' (QL:091), providing tenge 'to know', tengwe 'knowledge, understanding' etc.
* However, a root TEHE 'pull?' (QL:090, GL:069) is also of importance. It provides the stems for 'track, path, line' (tie, tê/tî), 'straight' (tína, tain < tegna), which persist to PE13:153 and survive to the Etymologies with the minimal change (perhaps, due to the importance of the 'Straight Road' concept to Tolkien).

In the Etymologies, the two stems were initially given as follows:

* 1937 Easier stem TEK 'make a mark, write, draw': in Quenya, tec 'write', tehta 'sign, diacritic', tekil < tekla 'pen'; tengwe < tekmē 'writing', tengwa 'letter' etc. Also tenkele 'spelling system', tekko 'stroke of pen'. In Noldorin the verb is teitha, 'mark' teith.
TEγ 'line, direction' initially given as a basis for Q tie 'path', N tê; téra, tîr 'straight, right'. Then the stem was altered to TEÑ (téra < teñrā), and at a very late stage the new form was re-inserted into its proper place (but with a very complicated reading and many alterations), changing its derivatives in the way into Q téma 'row, series, line', tea 'straight road'; N tî < tēñe 'line, row', tœr, taer < teñra 'straight'. Cf. malle téra lende númenna in the Lament of Atalante, featuring a different word for 'way' but the same as here for 'straight'. The addition of the new stem might be connected with the word indîw, innîw on the Moria gate drafts; Tolkien was already dissatisfied with TEK as its ultimate derivation.

This dissatisfaction (and a required alteration of all Quenya phonology concerning nasals in contact) can be traced through the whole of further changes. Consider WPP and its drafts:

* After 1955 The explanations of Moria gate. Verb teitha is derived from tektā in both draft and final source (PE17:043) and compared to Q tehta 'diacritic', S teith, taith same. Yet, the gloss of TEK 'make a written mark' does not exclude the possibility of remaining bare tec in Q, which is not discussed at all. The formula i thîw hin makes more complications: while in 1955 in a letter to Masson he still has no doubts that tengwa, tew are from tekmā, but as early as the draft an alternative TEÑ emerges (ibid.:044), first as a possible second derivation ('or both true!'), then the only possible. A reminder is put to alter the phonology such.
Characteristic of this process are the changes in the Outline of Phonology (see PE19:086): initially rewritten from OPD as such (favouring tengwa from TEK) and only then altered. However, as Tolkien noted already in WPP, the 'instrumental' suffix mā itself (cf. yulma) is not really fit to form a name for 'letter' ('_tekma_ would be a pen'), and yet tengwar are attested and must be accounted. So the alterations actually make not only km > kw but ñm > m with nasalization of the previous vowel! The only way out is now ñw, which is what Tolkien employs: ibid.:097.

* late 1950s This is the pattern observed in a carefully composed essay Quendi and Eldar:
(QE:394) "…the Loremasters, therefore, did not use lambe as a term for language or speech in general. Their terms were derived from the stem TEÑ 'indicate, signify', from which was formed the already well-known word teñwe > Q tengwe 'indication, sign, token'. From this they made the word tengwesta 'a system or code of signs'."

Therefore, the structure of the stems can be explained as following: TEÑ 'indicate' as the basis for tengwa (S têw) vs. TEK 'write' yielding tehta, taith. The remainder is given in the changes to the Outline of Phonology:

* "Owing to the disappearance of medial ñ the historical developments retaining a stem-nasal were specially liable to later alteration. Thus tēma, series, fr[om] TEG 'line': tegma, direct[ion], process > teñma > tēma. <…> The historical forms that survive were usually grammatically or etymologically isolated in the speech-feeling of Quenya-speakers: as tenna 'a thought, notion, idea' from base TEÑ; since[?] teñgwa, sign, indicator[?], letter is[?] from teñwā".

So, téma is explained here as a derivative of a different root, TEG 'line', obviously the same that produces (in hiatus) tegē > tie 'road'.

Now we shall observe whether the latest sources add to or contradict the established scheme somehow. Apparently, there is at least one attempt to rewrite this:
* 1969 PE22:149 repeats the same argument of TEK unavailable for tengwa and re-introduces TEÑ, but continues with some charts of phonetic development which would give tegma > tengwa. The word téma become inexplicable in this set-up (unless reassign both it and tie to a non-attested TE3, which is beyond the stretch of reconstruction).
* 1960s VT49:48 mentions a weak verb tengwa 'to read written matter', which is obviously derived from the noun and gives no additional information.

Significatio.

The situation described in QE and OP is most concise and carefully emerging, and I think PE22 additions can easily be passed on as an experimentation. A further confusion arises due to the connected (though barely intersecting) stems TEN 'direction, end', mostly (if not ever) invoked to explain tenna in Elendil's Oath, and strange TAN 'sign' homonymous to TAN 'construct' in Sindarin and apparently appearing only in a star's name, Tannacolli/Tengyl (the difference from TEÑ is doubtful, but can presumably be drawn as 'sign, symbol, vessel of meaning' vs. 'sign, token, message sent'?)

Next time: something concerning BA and WA and a reconstruction of a missing piece of 'Quendi and Eldar'.