According to legend, the name Czech is the name of the leader who settled the Czechs in their current homeland. The name of Bohemia comes from the Celtic Boii. It makes sense for a Slavic nation state to identify more closely with a Slavic name central to its origin story than with a Celtic one.

Oh boy I remember asking this question - https://www.reddit.com/r/AskHistorians/comments/1jxtdi/how_did_bohemia_turn_into_czechia_in_english/ - but I did not get a satisfactory answer sadly...

The thing is, the Kingdom of Bohemia did not just come close, it was quite literally "the Czech kingdom" for the Czechs. The Czechs/Bohemians and the Czech/Bohemian state always have been "Czech" (Češi/Česko/Čechy etc.) to Czechs themselves. Hence my confusion and questions.

I keep wondering when and how did the Czech - Bohemian switch happen in English?

And how did the situation in German influence it? With its Tsechien but also Böhmen?

The two terms in other than Czech have existed alongside for centuries - in fact they both are still in use! - so what happened?

As /u/peshun and /u/XWZUBU have both pointed out, "Czech" (Čech) was what the Czechs called themselves even while we in the United States referred to them as the Bohemians and the Moravians prior to the First World War (these terms are not outdated; "Bohemia" and "Moravia" are still the major divisions of the Czech Republic), though by the end of the war the term "Czech" would become common. As the Czech-American spy (and later CIA agent) Emanuel Voska noted in his memoirs of the First World War, hardly anyone in the United States knew at the time what a "Czech" was, and those that did assumed they were some kind of extremely foreign ethnicity, and not a peoples nestled up against the Germans. Voska, who joined the "Bohemian" freedom movement in the United States which called for Czechoslovak independence from the Austrian Empire, notes that they decided to call the freedom movement in the US the "Bohemian National Alliance" exactly because that was the name by which Americans knew them. This alliance (which led both the Czech and Slovak movements in the United States) was very influential in Europe, as it funded the operations of Thomas Garrigue Masaryk, the pro-Western Czechoslovak independence leader who would become Czechoslovakia's first president, though their effect in the US was nominal.

Once the United States entered the war, however, the views of Woodrow Wilson, who did not support Czechoslovak independence and in fact called for the protection of the Austrian state in his famous Fourteen Points speech, became a target for the Czechoslovak independence movement. During the final months of the war, Masaryk traveled to Washington, D.C., where he campaigned for Czechoslovak independence and met with Wilson several times. Aided by operations of the "Czech Legion" in Russia, which had originally traveled there as an ally of Russia only to become embroiled in the Russian Civil War, the term "Czech" became common parlance among the diplomatic elite in Washington (this is a rather complicated story). Secretary of State Lansing, for example, issued a memorandum to the media declaring that the aspirations of the "Czechs" and "Slovaks" had the "earnest sympathy of this Government" (Wilson did not yet approve). Finally, under pressure not only from Masaryk but also the Allied leaders in Europe, Wilson announced that he had been mistaken calling for Austria's protection and declared that the "Czechs and Slovaks" should decide what kind of government would suite them. Washington had begun to use the term "Czech," and this would never cease throughout the remainder of the twentieth century (sometimes incorrectly, as they would refer to all Czechoslovak policy as "Czech," even when referring to Slovak leaders such as the Prague Spring reformer Alexander Dubcek).

As evidenced by newspapers, however, the term "Czech" was not yet common among the general American population. But with the creation of a nation called Czechoslovakia, which was led by Czechs, the term "Czech" slowly crept into general American usage. The rise of Hitler and his demands that the "Sudetenland" of Czechoslovakia be transferred to Germany--one of the key events leading up to the Second World War--then thrusted the term "Czech" into the American mind. Because the conflict was between two clearly defined ethnicities, Germans and Czechs, there was no turning back to the days of "Bohemia." This time around, Czech propagandists in the United States chose to call themselves the "Czech Council" rather than the Bohemian Alliance, as they name Czech had become so commonplace. As you can see in this Google Ngram chart, the term "Czech" was more than 3 times as popular as "Bohemian" (which, remember, adopted new uses in the twentieth century) by the end of the Sudeten Crisis.

Sources:

Emenual Voska, *Spy and Counterspy*, 1940 (yes, it has an intruiging name, but it is a memoir, not a spy novel).

Betty Miller Unterberger, *The United States, Revolutionary Russia, and the Rise of Czechoslovakia*, Texas A&M University Press, 2000.

Gergory Ference (editor), *The Portrayal of Czechoslovakia in American Print Media, 1938-1989*, East European Monographs, 2006.