Chess historian Edward Winter has a page listing the earliest known occurrences of various chess terms in English, relying mostly on citations from the OED, though for these two terms he has slight improvements on those. For Zwischenzug the earliest known use is in Chess Strategy and Tactics, by Fred Reinfeld and Irving Chernev from 1933. For Zugzwang the earliest known use is in the February 1905 issue of Lasker's Chess Magazine.
As to why these terms caught on in English, I'd say the main factor, at least for Zugzwang, is that German is good at packing a lot of meaning into a single compound word in ways that English isn't. In fact, though Zugzwang was used as early as 1905 in English, Winter indicates in another article specifically on Zugzwang that usage of that term wasn't common in English chess literature until the appearance of the English translation of Nimzowitsch's My System. He notes that the translator, Philip Hereford, wrote in his introduction that Zugzwang was the only word he left untranslated
"mainly, because the single word conveys an idea, or complex of ideas, which can only be expressed in English by a circumlocution."
Sure enough, if you try to think of another way to say "Player A is in Zugzwang," you do find yourself needing to be much wordier, like "Player A is compelled to move, to his own disadvantage."
The same reason doesn't seem as important for Zwischenzug though, as it's no easier to call something a Zwischenzug than to call it an "in-between move," the literal translation of the term. I don't know, sometimes language is funny I guess.
Speculative update: Here's an off-the-wall guess. As Winter notes, the first use of Zwischenzug in English was in 1933. And he also thinks that the English edition of My System, which was shortly before in 1929, is what brought the use of Zugzwang into widespread use. So maybe that simply made it "fashionable" at the time to use a similar term like Zwischenzug as well. Chess players are already sort of like that, with openings coming into and out of fashion over time.