It seems that it was popularised by comedian Harry Enfield in his Loadsamoney sketch. The first newspaper references I can find – in the late 1980s – all refer to Harry Enfield, with the phrase gaining wider currency throughout the late nineties and into the 2000s (almost ten years in, I’m still not sure about the “noughties”).
Bish, bash, bosh - look at his dosh
There are earlier references in song titles (there is a jazz song “Bish Bash Bosh”, written by Barry Harris in 1962), and, according to The Guardian, in novels – a recent book review by Steven Poole states that the phrase “the bish-bash-bosh of freemasonry”, whatever that means, appears in a 1924 novel. I am not sure whether this means the phrase was in fairly wide use before Harry Enfield’s lighting on it, or whether he, like Barry Harris and the mysterious 1920s author, invented it independently of each other. The construction – three one-syllable nonsense words, with the noun changing from “i” to “a” to “o” – seems like a fairly intuitive one, if the examples above are anything to go by.
UPDATE - Dogberry points out that tic tac, mishmash, Kit Kat, knick knack, hiphop, clip clop and tick tock, ping pong, sing song, ding dong, and king kong all fit a truncated version of the pattern - an "i" followed by an "o" or an "a". So do flimflam, chit-chat, bric-a-brac (almost) and Mary Poppins' catchphrase "spit spot". Other phrases using all three vowels are "Slip, slap, slop" (as in "slip on a shirt, slap on a hat, and slop on some sunscreen") and Spike Milligan's On the Ning Nang Nong. There is a long list of similar words here - the "i" followed by "a" or "o" pattern seems remarkably prevalent.
Spit spot, and off we go...
This type of vowel alternation is known as ablaut, and the construction of phrases using vowels in this sequence is known as "ablaut reduplication" or "ablaut-motivated compounding". According to this grammatical essay, the "a", "i", "o"/"u" sequence is the most common because these three vowels are the the "fundamental pillars on which the whole system of vocalisation has been constructed", with "i" at one end of the spectrum (at the front of the mouth) "o" or "u" at the other (at the back of the mouth) and "a in the middle. The sequence is not just found in English - the essay cited above notes that phrases like "piff paff", "piff puff", "bim bam", and "bim bum" are also found in German.
Middle English had even more examples of irregular verbs using ablaut, like chide/chode, climb/clomb and help/holp. But, as this essay by a group of Harvard maths graduates reports, these have gradually died out and been replaced by regular forms, with the least frequently used disappearing first. The paper predicts that "wed" will be the next to go, with "wedded" replacing it as the past participle.

