As a little bit of background, the word 'chai' actually means 'tea'. Around most of Asia, a word sounding similar to 'chai' is used - in Japan and China, it is 'cha'.
When the Indians, Russians and Portuguese traded with China, they took up the usage of this word, which became 'chai'.
South Asian and Arab countries also have a version of this, using variations of 'chai' or 'shai'.
So how did we come to use the word 'tea' when the rest of the world (almost!) uses 'chai'?
In one province of Japan - the Fujian province - instead of using 'cha', they used the word 'te'. The Dutch traders who worked in that area, adopted that name and, with slight pronunciation changes, sold it as 'tay' through Europe.
Somewhere along the line, the English decided to change that pronunciation, and used 'tee' instead. And so we ended up with the way we pronounce it today!