In both Sindarin and Noldorin, any primitive final nasals [n] (and possibly also [m]) were lost at the end of words. Tolkien mentions this sound change several times:

In N. owing to earlier loss of t and final m, n in unaccented syllables... (PE21/58).

Since ns became s, while final n was usually lost in early Exilic [Noldorin]... (PE22/29).

For 5 Telerin had lepen, S. leben. In Telerin final n (< m, n) was not lost, but it was lost in Sindarin [emphasis added]; it is therefore probable that in Common Eldarin *lepen had assumed the form lepene with final vowel modeled on the other numerals (VT42/24).

The Telerin form [lepen] might go back to C.E. lepem with dissimilation of m: Common Eldarin -m survived as such in Telerin, but as n in Quenya and was lost in Sindarin (VT47/10).

One notable example of this change is S. and N. êl “star” vs. Q. elen from primitive ✶elen (PE17/67; WJ/360-3; Ety/EL). This example illustrates that the loss of final [n] was before the general loss of final vowels, because the second e was also lost. Furthermore, the plural form was S. elin < primitive ✶elenī, where the primitive plural suffix -ī prevented the n from becoming final and vanishing. Thus, these forms frequently had “irregular” plurals.

As was the case with the loss of other final consonants, the final [n] might sometimes have been restored by analogy with the plural. For example, in a few places Tolkien gave S. elen as the analogical singular for S. êl “star” (PE17/67, 139), and another example might be N. thoron vs. thôr “eagle”, plural therein (Ety/THOR), though in this case Tolkien actually stated that thoron was reformed from the archaic genitive ON. thoronen. If such reformations happened, they seem to be less common than restoration of other lost final consonants. In fact, plural variations like êl vs. elin seem to have been sufficiently common that they were generalized, and -in became a plural suffix appearing in places where it can’t be justified by historical phonology, such as conin plural of caun “prince” (LotR/953, PE17/102).

The loss of final m is complicated by the fact that Tolkien sometimes indicated that primitive final [m] became [n] in Common Eldarin (VT47/24, see note above). If so, there would have been no final [m] in the Sindarin period, and this rule can be limited to the loss of final [n]. But other times Tolkien seemed to think final m was preserved in Primitive Elvish (VT47/10, see note above), so perhaps it survived in Old Sindarin only to be lost in Sindarin. While there are plenty of examples of the loss of final n in Sindarin and Noldorin, finding clear examples of final m loss is difficult. David Salo suggested such a final m was lost from coe < ᴹ√KEM (GS/§4.54 and GS/243), and this is the only plausible example of such a change that I’ve found.

There are also a few examples where a final n seems to survive in Sindarin and Noldorin:

• S./N. aran “king” vs. Q. aran, both from √ARAN (PE17/147; Ety/ƷAR). See also short variant N. âr where the final n was lost (Ety/TĀ).
• S. rhovan “wild beast” vs. Q. hravan, both from primitive ✶srāban (PE17/78).
• S. talan “flet” (or N. talaf) vs. Q. talan, both from primitive ✶talam (PE17/52).
• N. coron “globe, ball” vs. ᴹQ. koron, from ᴹ√KOR (Ety/KOR).

These could represent reformations from the plural as noted above, or from slightly different primitive forms, such as *coronō vs. *coronŏ in the last example. A third possibility is that the loss of final n might not have been universal, but could be conditional based on stress patterns, as hinted at in the first note quoted above:

In N. owing to earlier loss of t and final m, n in unaccented syllables... (PE21/58).

It may be that the n was preserved when the final syllable was stressed, and that (for example) the primitive form of S./Q. talan was *talám. If so, then the loss of final n must have occurred before the stress shifted forward in Sindarin.

Conceptual Development: Given the lack of attested primitive forms for Gnomish words, it is hard to deduce whether this sound change applied at the earliest conceptual stages of the language. As suggested by Roman Rausch in his Historical Phonologies of Ilkorin, Telerin and Noldorin around 1923 (HPITN/§4.1.4), there are hints that final -n might have vanished in the Early Noldorin of the 1920s. The two forms he cited are:

• ᴱN. caifr “flea” vs. ᴱT. camparon, but its primitive form was given as ✶kamp’rū (PE14/66).
• ᴱN. ann “door”, plural ennyn vs. ᴱQ. andon, but its primitive form was given as ✶andond- (PE13/137).

As pointed out by Rausch, the evidence is ambiguous and there are other examples where n seems to be preserved: ᴱN. ailin “lake” vs. ᴱQ. ailin (PE13/136, 158), as opposed to N. oel (Ety/AY). Without further examples, it’s hard to say what the intended phonetic rules were.

The loss of final n seems to be solidly established by the Noldorin of the 1930s, though as discussed above when (and whether) this phonetic rule applied to final m is unclear.

See S. final nasals vanished after vowels for discussion.