In (Old) Sindarin and (Old) Noldorin, voiceless stops [p, t, k] became spirants before [s]. Tolkien described these changes for Old Noldorin in his notes on the use of the Feanorian Alphabet from the 1930s. The tengwar representations for the compendia characters in these quotes is approximate, since they are not supported in current tengwar fonts:

1+ was in older period used = [ts], but this group became [tþ], and so fell with tth from tt: written [13 stacked] (for 13): see compendia (PE22/27).

After the development of ON ts > tþ and the development of th generally to þ, the ON 1Ç = ts {became >>} now tþ and was given the shape 1Å = tþ, and from it were made z| = kth [with an up-hook not properly represented] (z| = ks [with a down-hook as shown]), qÅ = pth (q+ = ps). ps, ks having at the same time become phs, ꝑs, and χs were written e+ d| (PE22/28, plus note #90).

Based on these quotes, it seems that the first sound change was [ts] > [tθ], then later [ps], [ks] > [ɸs], [xs], and finally [tθ] > [θθ], the last aligned with the general trend whereby primitive [tt] > [ttʰ] (> [tθ]) > [θθ]. The sound changes [ps], [ks] > [ɸs], [xs] in Noldorin are obscured by the fact that the spirants [ɸ], [x] later became vowels and produced various diphthongs. There are several Noldorin examples of these sound changes appearing in The Etymologies, but you have to tease them apart from the later diphthongal developments:

• ᴹ✶tupsē [> *tuɸsa > *tousa] > N. taus “thatch” (Ety/TUP).
• ᴹ✶litse > ON. litthe > N. lith “sand” (Ety/LIT).
• ᴹ✶taksē [> *taχsa > *taisa] > N. taes “nail” (Ety/TAK).

From the first example, it seems the sound change [ps] > [ɸs] occurred after final [se] became [sa] (since the first example shows signs of a-affection), but the second example shows the sound change of [ts] > [tθ] took place before such e > a (and thus prevented [e] from becoming [a] since there was no [s]).

As pointed out to me by Bertrand Bellet at the Omentielva Toldea conference in August 2019, the Sindarin development seems to be different. Like Noldorin, there are a couple of Sindarin examples that show [ts] (> [θθ]) > [θ], but unlike Noldorin there is one example that strongly indicates that [ks] (> [xx]) > [x]:

• ✶lotse > S. loth “flower” (VT42/18).
• ✶otsōyā > S. othui “seventh” (VT42/25).
• √AKAS > aks[ē] [> *aχχe?] > S. ach “neck” vs. Q. akas, akse (PE17/92).

There are a couple of less obvious Sindarin examples that might also demonstrate [ks] > [xx] > [x]:

• S. carach “jaws, rows of teeth” (RC/607) might be the later form of N. caraes (Ety/KARAK) and thus cognate to Q. caraxë [karakse] as in Helcaraxë (S/134, Ety/KARAK).
• S. rasg “wain” is given as a cognate of Q. raxa in the Sindarin name of “Stonewain Valley”: Nan Gondresgion (PE17/28). But elsewhere this name is given as Imrath Gondraich (UTI/Stonewain Valley), and the second element raich might be derived from a variant primitive plural form *raksī (singular form *rach < *✶raksā), though Hammond and Scull reported this name as Imrath Gondraith (RC/558).

Also relevant is the fact that carach appeared as early as the 1940s in The Lord of the Rings drafts (SD/33) and there was variant form Helcharach for N. Helcharaes in The Etymologies of the 1930s (EtyAC/KARAK). Thus, Tolkien may have switched to [ks] > [x] as early as the 1940s and possibly even considered it in the 1930s.

There are no clear examples for the development of primitive [ps] in Sindarin, so it may have become [ɸs] (as in Noldorin) or [ɸɸ] (analogous to the Sindarin developments of [ks] and [ts]).

Conceptual Development: As noted by Roman Rausch in his Historical Phonology of Goldogrin (HSG/§2.5), it seems that the Gnomish sound changes were [ts] > [θ] and [ks] > [x] (and perhaps [ps] > [f]), consistent with the Sindarin phonetic developments:

• ᴱ✶aksa > G. acha “waterfall” (GL/17).
• ᴱ√TETE > G. teth “bud” vs. ᴱQ. tetse (GL/70, QL/92).

This was not a universal rule, however, since Tolkien said “s < ts after a long vowel” (GL/43); this was a factor in the phonetic development of gwais “kinship”, perhaps from primitive *ŋuaʒētse. As noted by Roman Rausch, it also seems that sometimes [ts] > [ss], perhaps under similar conditions:

• ᴱ√ROTO > G. ross “pipe” vs. ᴱQ. rotse (GL/65, QL/80).
• ᴱ√QETE > G. cwess “saying, proverb” vs. ᴱQ. -qet(se) “language” (GL/28, QL/77).

Finally, there is one Gnomish example where there seems to be Noldorin-style vocalizations of [xs] from [ks], but the relevant primitive form was revised, ı̯aksĕ >> ı̯aisa:

• First forms: ᴱ✶ı̯aksĕ [> *ı̯aχse?] > G. gais “steel” vs. ᴱQ. yakse (GL/37, QL/105).
• Revised to: ᴱ✶ı̯aisa > G. gais “steel” vs. ᴱQ. yaisa (GL/37).

As noted by Roman Rausch in his Historical Phonologies of Ilkorin, Telerin and Noldorin around 1923 (HSG/§4.1.3), in the Early Noldorin of the 1920s it seems that Noldorin-style vocalizations were the norm for primitive [ps], [ks], whereas [ts] > [θ] remained as in Gnomish:

• ᴱ✶kapse > ᴱN. cais “leap” (PE13/140).
• ᴱ✶lopse > ᴱN. laus [unglossed] (PE15/64).
• ᴱ✶orotse > ᴱN. oroth “impetus, speed, haste, rash courage” (PE13/151).
• ᴱ✶okswē > ᴱN. oif “terror, phantom” (PE13/151, 164).

But even in the 1920s, the vocalizations were not a universal rule, and there were still some Gnomish-style examples of [ks] > [x] and [ps] > [f]:

• [*tupse >] ᴱN. tuf “lump, knob” vs. ᴱQ. tupse (PE13/154).
• ᴱ✶ekse > ᴱN. ech “far away” (PE13/142).

Thus, it seems that [ts] > [θ] was part of Tolkien’s ideas from the very beginning and remained quite stable. In Gnomish [ps], [ks] (mostly) became simple spirants, but starting sometime in the 1920s they started to become vocalized spirants when followed by [s], though the development of the resulting diphthongs went through quite a few more conceptual changes after that. This remained the most common pattern through the 1930s. However, it seems that in the 1940s Tolkien switched back to the Gnomish pattern of ks > ch (whether this was also true of ps is unclear).

Neo-Sindarin: Tolkien’s vacillations in these phonetic developments is quite problematic for Neo-Sindarin, since adopting Tolkien’s later phonetic rules requires abandoning a number of well-known Noldorin words often used in Neo-Sindarin writing, like N. naes “tooth” and N. aes “meat”. It might be better to assume there were ancient dialectical variations that led to divergent phonetic developments, similar to proposed dialectical variations in the vocalizations of voiced and voiceless spirants in Neo-Sindarin.

HSG/§2.5