PE23/138 note #12 states this is a later change

In both Old Sindarin and Old Noldorin, voiceless stops [p, t, k] developed into spirants medially after nearly every other primitive consonant, in particular: all liquids [r, l], all nasals [m, n, ŋ], all aspirates [pʰ, tʰ, kʰ] and all other voiced and voiceless stops [b, d, g, p, t, k], that is essentially everything but [s]. In the last two cases, both stops in the pair became spirants, with various later vocalic effects (see below).

Some of these developments were the result of earlier sound changes in Primitive Elvish. First, primitive voiced stops unvoiced before voiceless consonants, after which aspiration moved to end of group of stops, so that primitive combinations of [b, d, g] with [p, t, k] became pairs of voiceless stops (e.g. [g] + [k] > [kk]), and the aspirate + voiceless stop combinations became voiceless stop + aspirate (e.g. [kʰ] + [k] > [kkʰ]). For combinations with liquids, nasals and voiceless stops, however, these sound changes occurred in (Old) Sindarin and (Old) Noldorin. A clear description of the origin of these sound changes for Sindarin appears in Tolkien’s Outline of Phonology from the early 1950s:

After other consonants p, t, k were generally maintained [in Quenya] though in fact (probably in C. Eldarin) after consonants other than [s], p, t, k were strengthened and lightly aspirated. (This development in Sindarin led to spirantalization as in rp > rf; but not in Valinorian dialects) (PE19/86).

Another clear description of these sound changes for Noldorin appears in Tolkien’s notes on the use of the Feanorian Alphabet in Old Noldorin from the 1930s:

A marked change of ON was the aspiration of t, p, k after all consonants. Since sp, st, sk and mp, nt, nk has special letters, this did not receive spelling-recognition in ON, except that in very late ON absolute initial [that is, at the beginning of a phrase] sp, st, sk Q, !, A became occasionally written e 3 d, i.e. ph, th, kh, or ee (eR), 33 (3R), dd (d$) indicating a change > sf, sþ, sχ > ff, þþ, χχ. This did not occur initially [that is, at the beginning of a word inside a phrase] (and here sp, st, sk remained in Exilic). r 4 f remained unchanged in use, and here aspiration probably did not proceed to spirantalization until the Exilic period. The letters Q are transcribed sp etc., and r mp without reference to exact phonetics (initial Q ! A are sometimes transcribed sph, sth, skh).

But in other cases [that is: medially] the aspiration became marked early, hence earlier ON [pt] q1, [tt] 11 or 1R, [kt] z1 and [pp] qq or qR, [kk] zz or zR are normally spelt as described above q3 etc. or compendiously qÅ. Hence in normal transcription pth, tth, kth, pph, kkh are used. Similarly in rth, lth (PE22/28).

Based on these two quotes it seems that (a) the medial spirantalization of voiced stops occurred fairly early in Sindarin/Noldorin phonetic history and (b) the mechanism was that the stops first became aspirates (a change which began in Common Eldarin), and then later the aspirates became voiceless spirants. These medial developments were distinct from the initial developments, where voiceless stops immediately became spirants after initial [s] without than first becoming aspirates.

At various points Tolkien discussed these sound changes in more detail for specific sets of combinations.

r, l + p, t, k: Tolkien discusses the phonetic developments of liquids + voiceless stops in several places:

The much changed Sindarin of Middle-earth turned the stops to spirants after l, r, as did Welsh: so *alkwā > alpa (Telerin) > S. alf (spelt alph in my transcription) (The Rivers and Beacon-hills of Gondor, late 1960s, VT42/7).

Old Sindarin (only preserved in Doriathrin records) retained intervoc[alic] m, and also the combinations mp, nt, ŋc in all positions; but p, t, c were already spirantalized after r, l except for lt [emphasis added]. At this stage thus appeared medial m; mp, nt, nc, lt; lph, lch, rth, rph, rch (notes on Sindarin dialectical developments, early 1960s, PE17/132).

As the second quote indicates, it seems the full change of [lt] > [ltʰ] > [lθ] took place later than the other combinations, which might help explain later phonetic developments unique to this combination: in particular for the restopping of final -lth to -lt in both Sindarin and Noldorin, and for medial lth becoming ll in Sindarin only. Examples of r, l + p, t, k producing spirants are numerous in both Sindarin and Noldorin:

[lp] > [lpʰ] > [lf]:	✶alkwā > [AT] alpa > [OS/ON] alpha > S. and N. alph “swan” (VT42/7, UT/265, Ety/ÁLAK).
[lt] > [ltʰ] > [lθ]:	malt + orn[ē] > OS. malthorn > malhorn > S. mallorn “gold tree” (PE17/50, VT42/27).
	ᴹ✶peltakse > ON. pelthaksa > N. pelthaes “pivot” (Ety/PEL, TAK).
[lk] > [lkʰ] > [lx]:	✶walka or ᴹ√(Ñ)GWAL > S. and N. balch “fierce, ferocious; cruel” (PE17/154, Ety/ÑGWAL).
[rp] > [rpʰ] > [rf]:	ar(a)- + pen > S. arphen “noble, knight” (WJ/376).
[rt] > [rtʰ] > [rθ]:	✶ṃbartā > ambarta > S. and N. am(m)arth “fate, doom” (PE17/104, 125; Ety/MBARAT).
[rk] > [rkʰ] > [rx]:	✶urkō or ᴹ✶orku > S. and N. orch “Orc” (WJ/390, Ety/ÓROK).

m, n, ŋ + p, t, k: Because nasals assimilated to following stops and aspirates in Primitive Elvish, the only possible combinations were mp, nt, ŋk > mph, nth, ŋkh (ŋch). Analyzing these changes is complicated by the fact that in both Sindarin and Noldorin later medial mf, nth, ŋch became mm, nn, ng, whereas final mf, nth, ŋch were restopped to mp, nt, nc ([ŋk]). Tolkien discussed the sound changes for nasals + voiceless stops for both Noldorin and Sindarin in several places, and in some later notes he said there were dialectical differences in the phonetic developments of Doriathrin and North Sindarin versus Ossriandic, West and South Sindarin:

Old Noldorin or Feanorian usage. Alphabet. After the letter is given its ON name in transcription; under the letter is given the usual transcription followed in square brackets by the phonetic value ... 10. r ampa: mp [mp, mph]; 11. 4 anta: nt [nt, nth]; 12. f anka: nk [ŋk, ŋkh] (Feanorian Alphabet, late 1930s, PE22/24).

r 4 f remained unchanged in use, and here aspiration probably did not proceed to spirantalization until the Exilic [Noldorin] period (ibid., PE22/28).

Similarly in oldest Sindarin, mp, nt, nc were preserved. In North Sindarin they universally became mf, nth, ŋch at the same time as lt, lp, lc and rt &c. > lth, lf, lch. The normal development in North Sindarin was then restopping when final > mp, nc, lt but nth, lf, lch, rth, rf, rch remained spirantal and medial nch > ¯ch. Later again nth > nt finally ... In Oss[riandic], West & South Sindarin changes proceeded as in North Sindarin far as mph, nth, ŋch. Then lt, mp, np, nc were restopped finally. Medially lþ, mph, nþ, ŋχ (though for long without change of spelling) became long voiceless l, m, n, ŋ. English transcription lh, mh, nh, ngh. In late Beleriandic Sindarin these became voiced ll, mm (m), nn, ng (= ŋg) unless followed still by a stressed syllable (notes on Sindarin dialectical developments, early 1960s, PE17/132).

Doriath, as might be expected from its history, was in general the most pure and conservative form of Sindarin in the Elder days ... The changes of mp, nt, ŋk also proceeded earlier and further than in the other dialects (ibid., PE17/133).

Based on the notes above, it seems that in both Noldorin and Sindarin, the spirantalization of voiceless stops after nasals took place later than the analogous changes after other consonants, with the possible exceptions of the North Sindarin and (maybe) Doriathric Sindarin dialects. As with [lθ], the lateness of these sound changes may help explain the eventual loss of the resulting spirants by them either restopping finally or becoming double-nasals medially. The obscuration of these later changes make it harder to find explicit examples of spirantalization of voiceless stops after nasals, but a few examples show the appropriate intermediate sound changes:

[mp] > [mpʰ] > [mf]:	✶ninkwiraite [> AT. *nimpiraite] > ni(m)phraed > S. niphred “pallor” (PE17/55).
[nt] > [ntʰ] > [nθ]:	ᴹ√THIN [> *thinta-] > ON. thintha- > N. thinna- “to fade” vs. ᴹQ. sinta- (Ety/THIN).
[ŋk] > [ŋkʰ] > [ŋx]:	✶dankĭna > daŋχen > danghen > S. dangen “slain” (PE17/133).

p, t, k + p, t, k: As with combinations of nasals with voiceless stops, the analysis of the spirants resulting from two adjacent voiceless stops is obscured by later sound changes. Tolkien described these sound changes several times, but mainly for double or “dynamic long” voiceless stops tt, pp, kk:

ON tt, pp, kk became tth, pph, kkh (Feanorian Alphabet, late 1930s, PE22/26).

Lengthening produced the following modifications: (i) p, t, k > pp, tt, kk. In the case of (rare) ph and of th, kh the original lengthening may have been pph, tth, kkh, but in Eldarin the aspiration was lost - or rather pp, pph became identical, either as pp in Quenya or as pph as in Old Noldorin [emphasis added] (second version of Tengwesta Qenderinwa [TQ2], circa 1950, PE18/91).

The long dynamic aspirates (Sc. formative lengthening of medial ph, th, kh) were probably rare in CE. In all Eldarin dialects they coalesced with voiceless stops, either by the loss of their aspiration on the one hand, or by the aspiration of the stops on the other. The latter occurred in Sindarin [emphasis added]. In Quenya the aspiration of pph etc. was reduced and these combinations did not become opened or spirantal (Outline of Phonology [OP2], early 1950s, PE19/87).

Of the other combinations, it seems only p, k + t ultimately remained. For t + p, k, the result was either transposed (tk > kt), or the t assimilated to the following stop to produce a double-consonant:

• ✶et-pel > eppel > S. and N. ephel “outer fence” (PE17/65, WR/137).
• ✶et-kat- [> *ekkat-] > S. echad- “outer fence” (PE17/65).
• ᴹ✶et-kelē > ᴹ✶ektele > S. and N. eithel “spring, well” (SA/eithel, Ety/KEL).

The long dynamic voiceless stops pp, tt, kk aspirated and became spirants ([ɸɸ] or [ff], [θθ], [xx]), and then later the long voiceless spirants shortened. Examples are plentiful: ✶rokkō > S. and N. roch “horse” (Let/282, Ety/ROK); ᴹ✶kwetta > S. and N. peth “word” (PE17/46, Ety/KWET); S. and N. ephel above. For the combinations [pt] and [kt], these became aspirated and spirantal as well ([ɸθ], [xθ]), but the first spirant in each pair [x], [ɸ] became a vowel before [θ] with various later effects. See the entry on the vocalization of such spirants for more details.

Conceptual Development: There is remarkable consistency in both Noldorin and Sindarin for these sound changes, but in the 1910s and 1920s the phonetic developments are necessarily different, because Tolkien did not introduce aspirates into Primitive Elvish until the 1930s. Prior to that point, the analogous set of phonemes in Primitive Elvish were themselves voiceless spirants (PE12/15). As noted by Roman Rausch (HPG/§2.2, §2.4; HPITN/§4.1.3), there are numerous examples of combinations with primitive voiceless stops which produce spirants in both Gnomish and Early Noldorin in the 1910s and 1920s. See the entry on how voiceless stops spirantalized after many consonants in Gnomish for more details.

GS/§4.64 @@@ mp, nc less clear

@@@ HSG/§2.2, 2.4, but maybe not mp: gwimpel