Again from www.worldwidewords.com, Quinion, his discussion of "tickety boo"

We can't be sure what its origin is. Eric Partridge always contended that
the word was forces' slang, most probably from the Royal Air Force, and
that it dates from the early 1920s or thereabouts (though the Oxford
English Dictionary doesn't find a written example before 1939).
Considering the number of Canadians who flew with the RAF during
World War II, its move to Canada isn't surprising.

The difficult bit is taking the word back any further than the 1920s. It could
combine that's the ticket - with much the same sense - with the childish
phrase peek-a-boo. But some find a link with the British Army in India,
suggesting it comes from the Hindi phrase tikai babu, which is translated
as "it's all right, sir".