Compare the following two Google Ngram Viewer charts for sceptical vs. skeptical in American English and British English:
My interpretation of these charts is that:
Before the 1910-1920's, sceptical was used in both American English and British English more than skeptical.
Something happened in the 1910's in American English that made the usage of skeptical become much more common than sceptical. This change never happened in British English.
Why did this change happen in the US? What was its trigger?
Note that while etymoline.com states that the sk- spelling is preferred in the US, it does not explain why it became so more common than the sc- spelling, in particular since the 1910-1920's.
skeptic also sceptic, (...) The sk- spelling is an early 17c. Greek revival and is preferred in U.S.