Actually, in English, Chai does not mean tea, it means a specific flavor of tea. If you don't believe me, try ordering some Earl Grey Chai, see what happens.
If your server is Indian, they'll likely react positively, and get you what you want.
Sure, but "chai tea" is still redundant. I have never used that term and ordered chai in many places without confusion.
https://www.google.com/maps/search/england+chai+london/