I don't think that Q. _nauka_, _nauta_ in the Lost Tales derive directly from NAQA, no; but it may bear "some connection" with the root of these forms (I know: vague much?). QL shows several related roots exhibiting a K/Q variation, e.g., PEKE > _pekte_ 'plume, comb (of cock)' and PEQE > _peqe-_ 'comb, card wool, tease', _peqen_ 'comb'; and KELU 'flow, run' > _kelu_ 'stream' and QEL + U > _qelu_ 'a well, spring, source'. So NAQA _could_ conceivably have a related form *NAKA (2) (distinct in meaning from attested NAKA 'bite'), and this does get us a lot closer to _nauka_ at least.
The U > AU is tricksier, yesss. There are a few examples in QL of roots in O having an extended form in OU, e.g. "_Laum-_, _Loum_, extended from _lom-_" > _laume_ 'a storm, overcast sky' (cp. LOMO- *'hide' > _lóme_ 'dusk, gloom, darkness'). We have no _explicit_ reference in QL to such a process taking place in
roots with A, although it might underlie the isolated form _laupe_ 'shirt, tunic', which was possibly derived via such a process from LAPA 'enfold', whence _lapa-_ 'wrap, swathe, wind' and _lapil_ 'a swathe, a flowing cloth'.
Be that as it may, applying Occam's Razor to the pair _nauka/nauta_ the simplest phonological explanation is a root NAWA or NAVA to which derivative suffixes _-ka_ and _-ta_ have been applied. NAWA is not recorded, unfortunately, unless we are to assume from the pair _nauto_, _nawa-_ in QL that _nawa-_ is the root of _nauto_ (which could be construed as the masc. form of _nauta_ 'dwarf'). If so, we are still left in the dark as to what NAWA meant, assuming it had some other meaning besides simply 'dwarf'.
BTW, it would be neat to derive _nau-_ from NU 'bow, bend down, stoop, sink', so that 'dwarf' was lit. 'stooped one', but this would require a-infixion, which was extremely common
by the time of the Etymologies (e.g., THUS- > *_thausâ_ > Q. _saura_ 'foul, evil-smelling, putrid', whence _Sauron_) but so far as I can tell is NOT present in QL. (L- and N-infixion are attested in the earliest period, but provide no help for the current problem.)
This leaves attested NAVA. Medial V in QL roots generally represents earlier "barred-B", a voiced bilabial fricative (to avoid going all Unicode on this list, I'll represent barred-B here with [B]). Notes placed with the Qenya Phonology state that [B] generally became V, but in the case of [B] + consonant became U (see Parma 12, pg. 23). With NAVA 'suspect, guess, have an inkling of', the development to V is seen intervocalically in _nâve_ 'shrewdness, sagacity, perspicacity' and _navillo_ 'whisper, hint suspicion, rumour, malicious comment, innuendo', whereas the preconsonantal development to U is seen in _naus_ 'suspicion', _nauma_ 'hint, clue', _nauta-_
'guess', etc. So NAVA doesn't present any phonological problems as the source of _nauka/nauta_ (it would be the source of _nau-_, with _-ka/-ta_ being derivative suffixes). The _semantic_ aspect, however, is less compelling, unless we are to suppose that _nauka/nauta_ 'dwarf' literally meant something like 'suspicious one', either indicating the untrusting nature of the Dwarves themselves or the opinion held of the Dwarves by those who had dealings with them.
If nothing else, I think the above demonstrates definitively why nobody wants to sit at the Linguist table at Mythcons.
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Thanks for the additional (and much more qualified :) thoughts, Pat. A question for you, relating to this comment:
> Andrew's supposition that there may be some connection with
> NAQA 'steal, take' is quit interesting, and
NAVA 'suspect, guess,
> have an inkling of' might also be connected, considering that some
> of its derivatives could apply to the negative portrait of Dwarves
> in the Tales,
e.g., _naus_ 'suspicion'.
Yes, interesting, I agree; but can you account for the change in both the vowel (a > au) and the second consonant (q and/or v > k)? Are these really close enough? A change of q > k is less of a leap in my mind, but v > k seems improbable at first glance.