abta- > apta-:
http://eldamo.org/content/words/word-1944249607.html

apta- > aptʰa-
http://eldamo.org/content/words/word-1683955225.html

aptʰa- > aɸθa-
http://eldamo.org/content/words/word-883570327.html

aɸθa- > auθa-
http://eldamo.org/content/words/word-4002924749.html

auθa- > oθa-
http://eldamo.org/content/words/word-567222053.html

In Primitive Elvish, the voiced stops b, d, g were unvoiced before suffixal t (PE18/102, PE19/90). Therefore primitively abtā would become aptā. I don't know of any specific examples of this development in Sindarin or Noldorin, though.

From there, I think the voiceless stops would have become spirants, and the p eventually vocalizing to u, as in S. leutha- from PE. lepta-

eldamo.org - Eldamo : Sindarin : leutha-

Another example of spirantalized p become a u is N. taus from PE. tupsē

eldamo.org - Eldamo : Noldorin : taus

Wait, sorry, I found another possible development. N. aes from PE. apsā and N. lhaes from PE. lapsā. It's possible that spirantilized p become i after front vowels and only u after back vowels. The evidence is ambiguous:

eldamo.org - Eldamo : Sindarin : [x], [ɸ] vocalized between a vowel and [s], [θ]

Based on leutha I believe that Tolkien changed his mind between Noldorin and Sindarin, so that the spirantalized p always became u, but that is extrapolated from a single example.

For some reason I didn't see +Ekin Gören posts as I was writing until I was done. I agree that his developments are the most likely, although aetha- is a possible result as well.

In Proto-Celtic and Brittonic, –bt– would > –pt– > –kt– > etc. (in the grammar that Tolkien read, at least). Indeed thank you both for taking the time to help and pointing out the difference in pre-Sindarin. (Eldamo is truly an Arda-phonologist's dream come true, once you know where to look.)

The devoicing in CE stage is actually described in TQ Mestanyatse, PE18:102, PE19:090.