In Sindarin and Noldorin it seems that medial [m] became [n] before the semi-vowel [j]. A similar sound change occurred in Welsh, when Primitive Indo-European -mi̯- became -ni̯- in Proto-Keltic (WGHC/§100iv). There are no published notes by Tolkien explicitly describing this sound change, but there are a couple Sindarin and Noldorin examples that demonstrate it:

• ᴹ✶ramya- > N. rhenio “fly, sail” (Ety/RAM).
• √MAY > amya- > anya- > S. ein- “excellent, admirable” vs. Q. amya- (PE17/163).

This sound change is the most plausible explanation of the medial n in the Noldorin form above, and the development is even more explicit in the Sindarin example. David Salo described this sound change in Gateway to Sindarin, but he attributed it to Ancient Telerin (GS/§4.31). Since there is no evidence of this sound change in Telerin, I prefer to be a bit more conservative and assume it is a Sindarin-only phonetic development, albeit an early one.

David Salo also suggested a similar sound change of [mw] > [nw], and there is a Noldorin example that illustrates this as well:

• ᴹ✶hamwa > N. hanw “?chair” vs. ᴹQ. hamma (EtyAC/KHAM).

However, an alternate form N. ham appears beside hanw, indicating Tolkien was not fully committed to this development. Furthermore, a note from the 1960s indicates that in Sindarin, Tolkien had [mw] > [mm] in at least some cases:

N.B. Final -w (left after loss of vowels) in Sindarin was dropped after labials (-mw > mm anyway) (PE17/148).

Tolkien may have reversed himself though, since in notes on the Common Eldarin Article (CEA) from 1969, Tolkien indicated mw > nw:

CE im wendēi should have produced phonetically S *inwendi with differentiation of mw > nw, a much favoured sequence in S. (as in Quenya), normally from CE nw and nm (PE23/139-140).

For now, my working theory is that [mj], [mw] > [nj], [nw] in Sindarin, but perhaps [-mw] > [-mm] when final.

See OS. medial [m] became [n] before [j], [w] for discussion.