To check the distribution of un-self-conscious usage of whilst by questioners and answerers at English Langauge & Usage, I ran a search for the word whilst in the EL&U search box. The search yielded slightly fewer than 600 matches. Then I excluded all instances that occurred in the context of discussions of while vs. whilst or other U.S. English vs. UK English topics. And finally (to the extent my memory allowed) I took only the first instance of whilst usage from any poster; in other words, I counted WS2 (for example) only once, even though he uses whilst often (and naturally).
My goal was to see how the usage of whilst broke out by country among posters who have identified their place of origin or current location in their site biography. To arbitrarily limit the size of the experiment, I decided to stop the counting when the number of posters whose home/residence was unstated reached 100. Overall, by the time I reached 100 unique whilst users of unknown residence, I had also tallied 85 users who had specified their place of origin or current residence.
Here is how the numbers came out for whilst users:
Unknown residence: 100
United Kingdom: 45
United States: 11
Czech Republic: 1
Hong Kong: 1
So on EL&U, according to the results of my experiment, a person who uses whilst in a non-whilst-related post is about four times as likely to live in the U.K. as to live in the United States. The numbers for whilst are undoubtedly higher proportionately in my experimental group than they would be in the wild (so to speak) among native U.S. English speakers. One of the U.S. matches, for example, came from a post-doc researcher at Texas A&M—and I know for a fact that no native of College Station, Texas, says whilst. As John Lawler observes in a comment beneath the question, use of whilst in spoken English by native U.S. English speakers is extremely rare.