They're using a transliteration that's usual for basically everything but English, the one that assumes common Latin sounds for all the individual letters, and then you just pronounce them one by one.
why the article contradicts me and says the tea is "chay" in Hindi and Urdu. I'm not saying the articles wrong, but maybe there's something I'm missing
Tranliteration, particularly into English, is fraught with myriad difficulties, the least of which being English orthography itself: there isn't a consistent mapping of sounds to letters. And there are often different systems of transliteration: from Japanese, 茶 (tea) can be transliterated as cha (Hepburn) or tya (Kunrei).
I don't read the article and the spellings as prescriptive with respect to pronunciation. I'm not familiar with transliteration of Hindi or Urdu (if someone more knowledgable on the subject would chime in, I'd love to learn a bit more), but I wouldn't read too much into the spellings.
