I don't understand the question. "Which day are we?" as a complete sentence makes no sense to me. I would say that in the U.S. we just don't use it.
Added: My wife (also a U.S. native speaker, from a different region) agrees. She's NEVER heard it.
However, it would be common and natural to say "Which day are we leaving on?"
Even Paul's locution, "Which day are we on?" sounds odd, except in an very unusual situation, almost but not quite contrived. Let's suppose some business had a weekday schedule, a Saturday schedule, and a Sunday schedule, and the important U.S. holiday, Thanksgiving Day, was coming up on Thursday. On Thursday, one might say "Which day are we on," meaning "I know we aren't on the regular weekday schedule, but are we on the Saturday schedule or the Sunday schedule?"
Added: My wife agrees. This was her example of a (rare) use of "Which day are we on?" Suppose you are following a day-by-day schedule of some program or task--a diet to prepare for a medical procedure, or a trip itinerary. You might ask "Which day are we on?" The answer would not be "Tuesday," it would be "This is day six, this is the day we are scheduled to visit visiting Devils Tower National Monument."