Chambers gives the following definition:
dated Brit colloq fine; satisfactory.
[predic.] British informal, dated
in good order; fine: everything is tickety-boo
Origin: 1930s: perhaps from Hindi ṭhīk hai 'all right'
I understand that the origin is India. In Hindi, "Tikai babu" means "It's alright, Sir." While India was governed under the British Empire, British airmen picked up the expression and it evolved into its current form. Canadian soldiers picked up the expression from their British allies during World War 2, and introduced it into Canadian jargon soon after.
On the other hand, Dictionary.com and TheFreeDictionary (both referencing the same source) suggest:
Origin: 1935–40; perhaps expressive alteration of the phrase that's the ticket.