Oddly, Eric Partridge, A Dictionary of Slang and Unconventional English, first edition (1937) gives an earlier date for buroo than Terry Victor & Tom Dalzell, The Concise New Partridge Slang Dictionary (2007—cited in Hugo's answer) does for "burroo, brew, buro." First, Partridge in 1937 directs readers of the entry for brew as a noun to buroo, and there he offers this entry:
buroo or brew. An employment-exchange : Public Works' coll[oquial] : from ca. 1924. I.e. bureau.
In the supplement to this dictionary (evidently added in 1937 or 1938), Partridge includes this entry:
buroo, on the. Out of work and drawing the dole: Glasgow coll[oquial] from ca. 1921. McArthur & Long. I.e. bureau.
"MacArthur & Long" refers to Alexander MacArthur & H. Kingsley Long, No Mean City: A Story of the Glasgow Slums (1935). A search of the contents of this book yields 15 matches for the phrase "on the buroo," including one on every odd-numbered page from 65 through 81, suggesting that the phrase appears in the title of a chapter running across those pages. At the very least, these matches indicate that "on the buroo" was in use in Glasgow by 1935. Nevertheless, for some reason, the Concise New Partridge doesn't venture a starting date for "burroo, brew, buro" of earlier than 1937.
An Elephind search for "on the buroo" in U.S. and Australian newspaper databases turns up an instance of the phrase from 1930. From "Topics of the Day: Highbrows on the 'Dole'," in the [Sydney, New South Wales] Catholic Press (October 30, 1930):
Bureau is, of course, the everyday French word for "office," writing desk, counting house, Government department, and many other meanings attached. It has long been adapted into the English speech. Since the British unemployment benefit, otherwise known as the "dole," was bestowed upon out-of-works the term has passed into the everyday vocabulary of the proletariat and so, according to what philologists and linguists call "the law of laziness," instead of using a circumlocution and saying "I'm drawing the unemployment benefit," the out-of-work with colloquial brevity prefers the concise word bureau, but with an accent and pronunciation unknown in Paris. He remarks "I'm on the buroo," or "I'm on the borrow," or "I'm on the barrow." In the Belfast Police Court recently a maintenance case was heard, and the defendant was asked by the magistrate what means he had to support a wife when he got married. "None," was the reply; "I was on the borough." Yet another shade of accent and pronunciation.
Use of buroo in the sense of Labor Bureau appears even earlier, in "Local and General News," in the [Coraki, New South Wales] Richmond River Herald and Northern Districts Advertiser (March 4, 1921):
The Unemployed.—The "Sun" tells a typical yarn: It was a warm, drowsy afternoon, and several hundred unemployed reclined on the grassy slope bordering the city's most dignified thoroughfare, while a small, perspiring person, with an amazing loquacity, harangued them fiercely. "Where can we go?" he howled. "We've been to the Premier, we've been to the Minister, we've been to the Labor Buroo. Where can we go?" A somnolent giant heaved his huge frame from the ground, stretched his mighty arms and yawned cavernously. 'You can go to ——," he remarked; "I'm going home."
Also, from "'Real Education': Work at Springfield School: Impressions of a British M.P." in the [Launceston, Tasmania] Examiner (November 17, 1932):
All their scholastic work showed character, and that character was the creation of the outdoor work on which they were engaged, which had instilled into them the saying of the Old Book, "Whatsoever thy hand findeth to do, do it with thy might." Not one of those boys or girls will ever be found at the "buroo" or in a queue of unemployed. Their natural instincts of industry have not been thwarted and destroyed and by the blending of the industrial part with the scholastic part, the earnestness of the one in which they delight is carried to the other which otherwise in most cases would have seemed to them insufferably boring.
A search of the British Newspaper Archive finds matches for "labour buroo" dating back to 1906. From "Hoisting of the Flag," in the Irish News and Belfast Morning News (August 27, 1906) [combined snippets]:
He repeated the statement to the objects of the "Belfast Labour Progressive Association" made at the previous gatherings, dwelling particularly on the establishment of what he called a "labour buroo" where men could have their names entered for a penny.
It also finds an instance of "buroo dole" from 1921. From "The Questions," in the Port-Glasgow [Renfrewshire] Express (September 30, 1921) [combined snippets]:
She continued by saying that she had never before been to a men's meeting in Port- Glasgow, but she could see that they were out for a merry evening. (Applause.) How was Robinson Crusoe able to live on a barren island without a buroo dole? (Laughter.) Colonel Greig: I suppose it was because he worked for his own living. (Hear, hear and applause.)
And it finds multiple instances of "on the Buroo" from as early as 1922. From an unidentified article in the Glasgow [Lanarkshire] Forward (May 20, 1922) [combined snippets]:
Scene: a quiet corner where millers are pitch-and-toss for bawbees.
James[?] who has gone broke: Gie's o' tuppence, Charlie.
Charlie: Hoo in heaven's name dae [ex]pect I can lend ye tuppence workin' every day--ask Peter the he's on the Buroo!
The Glasgow Forward again uses the phrase on September 2, 1922 ("And when their daring deeds are done They---starve on the Bur—oo.") and yet again on September 16, 1922 ("and the Beard crowd haven't just been so flush, still they're not yet on the Buroo"). It thus appears that the old Partridge dates for "buroo" ("ca. 1924") and "on the buroo" ("ca. 1921") were much closer to the actual starting dates for these terms than the later Concise New Partridge Dictionary's date for "burroo, brew, buro" ("1937") would suggest.