+Leonard W. Well, the roster of Sindarin derivational suffixes with unclear meaning is quite huge (above Quenya's), and nathla seem to be quite legit judging by phonology and comparing to nathal 'guest'. However, I'm in doubt for two reasons:
• When the basic meaning of stem is verbal, a basic verb is expected, so just nath is what I'd prefer;
• Quenya has an attested verb, preferred to mimic the English word 'well-come' as alatulya 'well-lead', though it is crossed out and obsoleted by reaasignment of ala to 'not'; something like maitulya, S. *maedelia might be still an option.

and for some reasoI cannot place I find a mention in my notes: 'NATH: unsafe root. Obsoleted?' If I can remember my reasoning behind that I'll share it.

I think that the ancient root is a noun, NÁTHAL. For nathla-, the ancient derivation was probably NÁTHAL+AA, (one way to make a verb out of a noun was to just add a long A to the end of it) which became nath'la in Sindarin. Quenya equivalents would be Nasal and ... help me out, Quenya Ranglers. The first few steps are easy enough: natʰala: > natʰla:... then would it become naltʰa:? or not change at that point? In Helge's article about Quenya's phonological development, he mentioned that sometimes the syncope wouldn't happen if an undesirable consonant cluster would result (the example tatʰare is given). Perhaps /tʰl/ is an example of one such undesirable consonant cluster?  So, it would stay natʰala:. The the next step would be naþala-, and later on in Exilic Quenya: nasala-.

I think most likely a primitive form nathalā would undergo the Quenya syncope to nathlā. From there, according to PE19/88, the thl would sometimes undergo metathesis to lt, so the end result would be nalta-.

If not, the resulting thl > þl > þal, eventually producing nasala- as you suggested (again according to PE19/88).

Note that for Helge's conjecture that the syncope was prevented for undesirable clusters, according to PE19/88 the combination thl always became þal, so the development could have been tathare > tathre > taþre > taþare > tasare.

But isn't it just as possible that the root is NATH as Tolkien gives it in the text and nathal is formed along the same lines as Q hatal from HAT-?

+Fiona Jallings Tolkien's own postulation about the stem being NATH would rather contradict your conjecture; also, my very limited knowledge of the súndocarmë is that the final consonant is normally a stop or [s] (and rarely [r]); hence, such an extension does not seem really probable to me.