Is this the anglicised version of what the French say after someone has sneezed which is:- "A tes souhaits" or God Bless you? If this is said quickly sounds like atishoo and the French do say it quickly after someone has sneezed.
- This sounds plausible to me, but unprovable.
I just heard a French person reply "A Tes Vous" to a sneeze and made the same connection. I do know that different languages have different "Sneeze sounds" and deaf people don't have "Sneeze sounds" when they sneeze.
Source: http://www.bbc.com/news/blogs-ouch-23162903 Also English does have a history of mangling other languages into loan words.