They do have Federation Standard as a language, which I assume is probably English with lots of words from Vulcan, Andorian, and Tellarite, but not everyone knows it. The problem in that episode though wasn't that the UT was down, it's that it was translating what everyone was saying, or reading on consoles, into other languages. I'm sure there were places on the ship we didn't see, where some historian was so excited to have a use for their pen and paper, so they could communicate through written Federation Standard lol