Based on the example ✶rinsa > S. riss “cleft, cloven, separate” (PE17/87), it seems that [ns] became [ss] in Sindarin and Noldorin. In the history of Welsh, -ms-, -ns- > -ss- in Proto-Keltic (WGHC/§95iii), so Tolkien could have modeled the sound change after that. Some 1957 notes on nasal mutation indicates this sound change was a factor in the (archaic) nasal mutation of voiceless lh, rh from primitive initial sl, sl:

Late forms as nan-h as for vowels, archaic nath-r, nath-l for nan-rh, nan-lh. nan|sr > nassr > nathr (PE17/147).

It was probably also a factor in the nasal mutation for s [in·s > is·s > i·s]:

• S. Echad i Sedryn “Camp of the Faithful” (UT/153).

Tolkien mentioned this sound change in notes on the Noldorin usage of the Feanorian Alphabet from the 1940s, though he seems to have been uncertain of the exact development:

Since ns became {ss >>} s, while final n was usually lost in early Exilic [Noldorin]... (PE17/147).

There is another mention of this sound change in notes on Quenya verbs from the 1940s, attributing ns > ss to Primitive Elvish:

s. This became [z] medially; but ns became prehistorically (common to Noldorin and Quenya) > ss (PE22/103).

Tolkien mentioned this change in notes on the Common Eldarin Article (CEA) from 1969, where he attributed it to Ancient Telerin:

A still earlier change common to T. and S. was ns > ss (the preceding vowel remained unchanged). This loss of the nasal (by unvoicing and assimilation at a time when s was the only voiceless spirantal sound) also occurred before sl-, sr-, but before a further consonant the s was not doubled (PE23/103).

There was a similar (and better documented) sound change in Ancient Quenya, which further supports the idea that this phonetic development could have occurred in Common Eldarin, but the fact that this sound change was a factor in Sindarin nasal mutation points to a later (and parallel) phonetic development.

In Quenya (and Welsh) ms > ns > ss as well. In CEA the plural definite article was derived from im and the result was still nasal-loss, so it seems like ms > ns > ss also applied to Sindarin.

Conceptual Development: There are couple examples of ns in isolated names from Noldorin and Gnomish: N. Pensarn (TI/366) and G. Bansil (LT2/214), which could mean this sound change does not apply to later compounds in Tolkien’s earlier writings. However, the name S. Arassuil (= Aran + suil?) may be an example of a late compound from the 1950s-60s with this mutation. Hat-tip to Parmandil for this example.

Neo-Sindarin: Given Arassuil and the behavior of the nasal mutation before s, it is probably safest to assume that ns > ss remains an active sound change in Sindarin.