Tea didn't really spread to the west over time or even that long ago. It also didn't spread from China to India to the Middle East to Europe, or at least it didn't catch on in any significant way..

..until in the colonial era European traders go to China and bring back tea to Europe, and in the 17th century the British catch on to it and develop a craze for it. They import massive amounts from China and in an effort to produce their own, cheaper tea, they smuggle tea plants out of China and start plantations in India.

And then tea was introduced into the Middle East through trade with Europeans, and to Persia from India by land. In Turkey, where they drink the most tea in the world per capita, tea caught on as a cheaper alternative to coffee when there were shortages and it got expensive in WW1.

tl:dr; it spread from China > Europe > India > Middle East through trade in the colonial era.