The second usage citation (for tickety-boo) from the OED2 is of interest: 1947 Amer. N. & Q. Sept. 94/1 Lord Mountbatten, now Governor General of India, is credited in the New York Times Magazine (June 22, 1947, p. 45) with ‘giving currency’ to the phrase ‘tickety-boo’ (or ‘tiggerty-boo’). This Royal Navy term for ‘okay’ is derived from the Hindustani.”
Not familiar in the US. I think I first encountered it in The Hitchhiker’s Guide to the Galaxy (original radio show, 2nd season IIRC): “Hi there guys! This is Eddie your shipboard computer welcoming you back on board the starship Heart of Gold. We are currently heading away from the planet Earth on improbability drive, and all systems are just tickety-boo.”