I will advise everyone who speaks a foreign language to speak fairly loud to minimise the risk of misunderstandings. Some people feel shy and uncomfortable when speaking in a foreign language, which unconsciously makes them lower their voice. Also, make sure you have eye contact. It’s often possible to tell if someone doesn’t understand you by observing their facial expression. If you’re not sure if you were understood, then ask for confirmation. Say for instance ”I wish to be sure you understood me correctly. Could you please tell me what you think I said?”

That's a very strange story, and an unusual experience. I have two guesses, but they are only guesses. The first is that in the United States, jobs in places like Dunkin' Donuts are sometimes held by recent U.S. immigrants with limited English skills. It may have been a case of difficulty with English on both sides of the conversation--neither could understand the other one's accent. These workers would probably recognize phrases like "an everything bagel" or "a strawberry Coolatta," but not "a cup of tea." I think I have it! A very common order at a coffee shop would be "a Cappuccino." If the student said "a cup of tea" it is (barely!) possible to imagine hearing "cuppaTEE" as "cah-pa-CHEE," i.e. a Cappuccino. A Cappuccino is is a brown-colored sweet coffee drink made milk, cinnamon, and foam. (It's an Italian name, named for the brown robes worn by Capuchin monks). A second possibility is that they might not have had tea! Or, tea orders might be so rare that the server didn't know that they had tea or where to get it. This, too, is surprising. Most places at least know how to give you a cup of hot water and a tea bag, and most of the "gourmet coffee" chains now offer some "gourmet" teas as well. As a bit of travel information, I will say that in the last few years in the U.S. we have just started to see the rise of specialty tea shops (distinct from coffee shops). I see them in malls and on trendy shopping streets, and if you want tea you might have a better time looking for a chain like "Teavana" or "David's Tea" then at any coffee place. You can get many kinds of tea there (Lapsang Souchang, etc.) but whether they can make ordinary British-style tea, I don't know.

Interesting discussion. Dan’s theory that the employee may have been a nonnative speaker is quite possible. I also agree that “cup of,” while usual in ordering tea in the UK, would be less commonly said in a typical US coffee shop, and the /ʌ/ could easily be mispronounced by non-native English speakers. As far as confusing the first consonant in “tea” (/t/) with /tʃ/, this is the usual pronunciation of a T before /i/ in Brazilian Portuguese, a language with hundreds of millions of speakers. There are other major languages with the same issue, such as Japanese. To some extent, even the palatalized T use in Slavic languages could cause confusion. An additional factor may be that some RP speakers aspirate the T (especially in initial accented position) so much that it may be confused for an affricate /ts/ — this would sound almost like the first letter in cinnamon. (Also typical of Canadian French before /i/.) Perhaps the student was imitating this pronunciation and went a bit overboard. What is the student’s native language anyhow?

I suspect that the assistant didn't even hear the word 'tea'. The problem probably arose because the assistant picked up on the syllables 'cup of ...' and presumed the student had said 'cappuccino', as Dan Smith suggests.

Impossible to know without hearing her say it. "T", how can it be mispronounced?