MMario had the "accepted" explanations of tickety-boo at the start of this thread. Lord Mountbatten is credited in the NY Times Mag. (1947) with giving currency to the phrase tickety-boo (or tiggerty-boo). The Royal Navy term for okay is derived from the Hindustani (Hindi thik hai, all right; babu, sir). Streatfeild in 1939 was first to put the spelling 'tickety-boo' in print.
I still think the Danny Kaye song was the one that put the term in American minds. In Canada, a WW2 veteran I called said that it was armed forces language.

Reminds me of the argument about "Bless 'Em All," American 1940 vs. "Fuck 'Em All," anecdotally British Army in India, about WW1 time.

Hunky was U. S. slang in Civil War time (1861) for safe and sound, all right. No one seems to know where the 'dory' came from. Hunky-doree appeared in print in 1866 (U. S.). H. L. Mencken considered it an artificial word (The American Language, 4th Ed., p. 145)