The colloquial Australian-English phrase not to know whether it’s Tuesday or Bourke Street, and its variants, are used of a state of confusion or stupidity.

The following explanations are from Similes and other evaluative idioms in Australian English, by Pam Peters, published in Phraseology and Culture in English (Berlin: Mouton de Gruyter,  2007), edited by Paul Skandera:

Urban street names appear in postwar idioms for confusion or being flustered: doesn’t know whether it’s Tuesday or Bourke Street (Melbourne), which becomes doesn’t know whether it’s Pitt Street or Christmas (Sydney)—or doesn’t know if it’s Thursday or Anthony Horderns, a reference to a former Sydney department store. The relexification of this originally Melbourne formula in Sydney is again a sign of the lively connections between phraseology and Australian oral culture—and of the endemic rivalry between Sydney and Melbourne! The names of the dominant local retailers are enshrined also in the similes more front than Myers and more front than Foy and Gibsons, where the play on “front” makes it a comment on someone’s impudence or bravado. This is also the point of the Sydney-siders’ simile more hide than Jessie, an allusion to a much-loved, long-lived elephant at the Sydney zoo, which puns on the word hide with its Australian colloquial meaning ‘impudence’.
Yet another metropolitan idiom, gone to Gowings (referring to a men’s department store in Sydney) is remarkable for its polysemy—or indeterminacy of meaning.

The earliest occurrences that I have found of the phrase not to know whether it’s Tuesday or Bourke Street and variants are as follows:

1-: From The Ridge and the River (1952), a World-War-II novel by the Australian author Tom Hungerford (1915-2011)—as published in The Sunday Herald (Sydney, New South Wales, Australia) of Sunday 22^nd June 1952:

“Womai!” Shearwood beckoned him. “This place, Tononoi. You got savvy ’long ’im?”
The kanaka shook his head briefly, and Malise made a sharp movement of impatience.
“You waste too much time on the dope. He don’t know whether it’s Tuesday or Bourke Street. I’d give him a good stow in the tail and send him back to the bloody Japs, that’s what I’d do—or clap a bag of tins on his back, the bludger!”

2-: From Josh (Sydney: Angus and Robertson (Publishers) Pty. Ltd., 1971), a novel by the Australian author Ivan Southall (1921-2008)—as reprinted in 1988 by Macmillan Publishing Company, New York:

“Oh, leave him alone, Harry.” Betsy sighing. “I told you he was dumb. He doesn’t know whether it’s Bourke Street or Friday. He’s harmless.”

However, in A Dictionary of Australian Colloquialisms (Sydney University Press in association with Oxford University Press Australia, 1990), s.v. not to know whether it’s Tuesday or Bourke Street, Gerald Alfred Wilkes (1927-2020) quoted this passage from Ivan Southall’s Josh as follows:

‘I told you he was dumb. He doesn’t know whether it’s Tuesday or Bourke Street.’

—Cf. also:
– ‘back o’ Bourke’: meaning and origin;
– ‘things are crook in Tallarook’: meaning and origin.

The synonymous phrase not to know whether it’s Pitt Street or Christmas occurred, for example, in Come In Spinner (London: William Heinemann Ltd, 1951), a novel set in Sydney at the end of the Second World War, by the Australian author Dymphna Cusack (1902-1981) and the New Zealand-born Australian author Florence James (1902-1993):

Elvira looked up at the clock. “’Oly Moses,” she squealed, “I ’ad to take coffee to old Ma Dalgety at nine—black and strong. Well, I’ll tell ’er it’s nine. She’s that pie-eyed, she don’t know whether it’s Pitt Street or Christmas, and ’er diamond watch don’t go since she dropped it into a glass of gin with ’er false teeth the other night.”

The synonymous phrase not to know if it’s Thursday or Anthony Horderns occurred, for example, in Lily on the Dustbin: Slang of Australian Women and Families (Penguin Books Australia Ltd., 1982), by the Australian author Nancy Keesing (1923-1993):

More to the point is her brother’s remark that she’d ‘forget her head if it wasn’t screwed on’. Her aunt sums up: ‘You wouldn’t know if it was Thursday or Anthony Horderns’.