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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.

No, just imitative of the sound that a sneeze makes. Equinox 14:42, 1 June 2015 (UTC)
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: Also English does have a history of mangling other languages into loan words.