Representing local pronunciations, buroo, broo, b’roo and brew are informal Scottish and Irish forms of the noun bureau.

The earliest instance that I have found predates by twenty years the one recorded in the Oxford English Dictionary (OED – 2^nd edition, 1989), and is used in the generic sense of office, agency—not in either of the two specific senses defined below (the only ones mentioned in the OED). It is from the column By the Way, in the Berwick Advertiser (Berwick-upon-Tweed*, Northumberland, England) of Friday 18^th September 1914:

The Press “Buroo.”
A correspondent sends me the following tale, which I give with all reserve:—In a certain little shop a number of workmen were sitting discussing the war in its various phases. The subject of war correspondents cropped up and one of the company contended that there were no war correspondents allowed at the front. Another man agreed with the previous speaker to a certain extent. He argued that all war correspondents were excluded except one and “he signed on before the war started. They ca’ him Press Buroo or something like that!”

(* Berwick-upon-Tweed is a town at the mouth of the River Tweed in north-eastern England, close to the Scottish border; having been held alternately by Scotland and England, it was ceded to England by Scotland in 1482.)

The Scottish and Irish forms such as buroo have come to specifically denote a Labour Bureau, i.e. a government office from which unemployment benefit is distributed, hence also the unemployment benefit itself.

The following illustrates those later meanings; it is from an account of the trial of James Inglis and Patrick Docherty, charged with having stolen clothing, published in The Falkirk Herald and Scottish Midlands Journal (Falkirk, Stirlingshire, Scotland) of Saturday 25^th February 1922:

Inglis, who gave evidence for the defence, said he had been drinking all that morning and all the night before and could not remember whether or not Docherty was with him at the time the offence was committed. Docherty had a few shillings left from his “buroo” money, and on the Friday night he helped him to dispose of it.

The earliest instance that I have found of the informal phrase on the buroo, meaning on the dole, on unemployment benefit, is from The Sunday Post (Glasgow, Lanarkshire, Scotland) of Sunday 23^rd October 1921:

FARES, PLEASE!
SEEING LIFE ON A GLASGOW TRAMCAR.
BY A CONDUCTOR.

I am often asked what is the finest tramway service in the world, and I admit Glasgow.
Here am I, dressed up in a uniform of green, strapped across my breast is a large piece of polished metal, similar to that very serviceable article a Chubb lock. Inside this contraption is [a] mechanism which punches a hole in the ticket and at the same time rings a bell.
As you punch a ticket it is registered and the number is shown on the plate. If the bell is silent get hold of an inspector and report this fault. This is a hint—never let a chance pass of reporting to these “referees.” This is only getting a little of your own back.
I also carry another article across my shoulder, and this article is the cause of envious eyes—the money bag. As per Rule No. 1, this bag must never hold less than 3s 6d change. You get this cash along with the bag, and heaven help you if you ever feel tempted to spend twopence of it. If this happens, and a human “X-Ray” discovers it—well, you are on the “Buroo” for a spell.
All the fares you receive go into it, and the weight sometimes nearly strangles you. How to prevent it getting too heavy is to live in hope of someone asking for a 1d ticket and tendering a 10s note. Then is the chance to get rid of fourteen pounds of metal.

The second-earliest occurrence of on the buroo that I have found is from the Aberdeen Daily Journal (Aberdeen, Aberdeenshire, Scotland) of Wednesday 29^th March 1922, which reported that a house tenant, who appeared at the Aberdeen Sheriff Court for not paying his rent, declared that

He could pay nothing at present as he was unemployed and on the “Buroo.”

The earliest Irish use of on the buroo that I have found is from the Belfast Telegraph (Belfast, Antrim, Ireland) of Tuesday 5^th February 1924:

UNEMPLOYMENT BENEFIT.
CHARGES OF MISREPRESENTATION.

At the Belfast Summons Court, before Mr. James Roche, R.M., and other magistrates, a number of summonses for making false representation to the Ministry of Labour, and receiving unemployment benefit, were heard.
John Brown, of Riverdale Street, was summoned for making a false representation that he was not in receipt of a disablement pension on days between April, 1922, and March, 1923.
The Secretary of the Orange and Protestant Society said defendant had received disablement benefit during the period in question from the Society.
The defendant said he was in receipt of a small pension of 8s per week from the Army. He had committed this offence through ignorance of the law. He was 61 years of age and had gone through the South African and Great War. As soon as he discovered his offence he signed off and was not on the “buroo” at present.
In reply to Mr. Roche—He was willing to pay the money back.
The case was adjourned for two months to enable defendant to pay back one-third of the money received.