How has it happened and Kemal Ataturk did not adopt Greek Alphabet, although in the Ottoman empire the Greek (and Cyrillic) were spoken?

There was use of Greek script to write Turkish: Karamanli Turkish. Illustrated in https://www.quora.com/How-has-it…

But without some concerted linguistic  work, Greek script was not much better suited to Turkish than Arabic script was. No differentiation between <ı> and <u> for example: both ου. No systematic differentiation of <c> and <ç>, just as Greek (at the time) did not differentiate /ts/ and /dz/: both τζ. No smooth way of rendering <ö> and <ü>. No differentiation of <ş> and <s>.

Now these were not insurmountable difficulties. The Soviet orthography for Pontic (now reborn on the Pontic Wikipedia) dealt with similar issues by inventive use of digraph, and by ignoring any backward compatibility with metropolitan Greek.

But why bother when

• there was a lot more precedent of using Latin for new language scripts (Cyrillic was mostly a Soviet-era thing), including diacritics (which only Greek dialectologists use)
• Latin was consistent with Kemal’s imperative of Westernisation
• Latin did not have any significant associations with minorities (Levantines were few, and seen as representatives of the West anyway)

There was also the small issue of who the Turkish War of Independence was fought against…