As a young American, I can say that I have no recollection of his name. I've never been particularly interested in space exploration, but living in Florida, we visited the Space Center, watched launches, etc. The names I know are all American: Neil Armstrong, Buzz Aldrin, etc. The "programs" I know are too: Apollo, Challenger, etc.
We were taught that the Russians made it to space before we did, and I think that's about it before we moved on to American achievements. Perhaps I had a nationalistic teacher.