I should add, however, that the big irony in all of it is that if Turkey wanted to around the early stages of independence, it would have been more or less easy to adapt Standard Turkish, on the basis of the İstanbul dialect, to the Cyrillic script as adapted for Russian post-1918. With a few modifications to match Turkish phonology to the equivalent Russian pronunciation, plus some additional letters (three from the OTL Azeri Cyrillic alphabet, one from Serbian/Macedonian, another from an old 19th-century proposal for representing foreign loanwords in Russian, and another two which in this case are specific to Turkish), one could have the following (Cyrillic to the left, OTL Turkish plus additional explanations for context on the right):
А, а = A, a
Б, б = B, b
В, в = V, v
Г, г = G, g or Ğ, ğ
Ғ, ғ = G, g (in foreign borrowings and proper nouns, mainly before back vowels)
Д, д = D, d
Е, е = Ye, ye (word-initial and after vowels); E, e (after consonants)
Ё, ё = Yo, yo (word-initial and after vowels); Ö, ö (after consonants)
Ж, ж = J, j
З, з = Z, z
И, и = İ, i
Й, й = Y, y
К, к = K, k
Ҝ, ҝ = K, k (in foreign borrowings and proper nouns, mainly before back vowels)
Қ, қ = K, k (etymological */q/ from Arabic loanwords)
Л, л = L, l
Љ, љ = L, l (in foreign borrowings and proper nouns, mainly before back vowels)
М, м = M, m
Н, н = N, n
О, о = O, o
П, п = P, p
Р, р = R, r
С, с = S, s
Т, т = T, t
У, у = U, u
Ӱ, ӱ = Ü, ü (word-initial and after vowels)
Ф, ф = F, f
Х, х = H, h
Ц, ц = Ts, ts (confined to Russian loanwords, and most likely pronounced <S/s> anyway when word-initial due to a prohibition of word-initial consonant clusters in Turkish)
Ч, ч = Ç, ç
Ҹ, ҹ = C, c
Ш, ш = Ş, ş
Щ, щ = Şç, şç (mainly in Russian loanwords)
Ы, ы = I, ı
Ь, ь = silent (used to separate /j/ from other readings when a iotated vowel is after a consonant)
Э, э = E, e (word-initial and after vowels)
Ӭ, ӭ = Ö, ö (word-initial and after vowels)
Ю, ю = Yu, yu (word-initial and after vowels); Ü, ü (after consonants)
Я, я = Ya, ya (word-initial and after vowels); Ä, ä (confined to Russian loanwords and pronounced identically to <Е/е> after consonants)
’ = Y, y (mainly in Russian loanwords where <ъ> would be expected; Bulgarian, OCS, and other loanwords with schwa would probably follow the Moldovan Cyrillic alphabet and use <Э/э>, while Russian loanwords with schwa would be spelled as per Russian orthographic principles, though preferring <А/а>)
Most of it is pretty straightforward and wouldn't prove a problem. <Қ, қ> is used here because there are some Arabic loanwords in Turkish which violate vowel harmony due to its origin in */q/. Also, in native Turkic words, <к, г, л> automatically soften when before a front vowel; as the soft sign here is reassigned elsewhere (leading towards what one would probably stereotypically call a "Tatar accent"), the additional letters <ҝ, ғ, љ> highlight those exceptions where soften consonants occur outside of their natural environment, particularly in foreign loanwords. As for <г> - it represents <g> when word-initial or geminate and <ğ> otherwise (since the latter cannot appear at the beginning of a word in Turkish).
So there - if someone decides to go for a Communist Turkey TL, here's a Cyrillic alphabet you can use.