In British English, the expression old boy has been used to designate:
– a male former pupil of a particular public school (i.e., a private independent fee-paying secondary school);
– favouritism existing among former pupils of public schools, or other old friends.

In particular, the phrase old boy network (also old boys’ network, old boy net, old boys’ net) designates a system of favouritism and preferment operating among people of a similar social, usually privileged, background, especially among former pupils of public schools.

—Cf. also the phrase old school tie.

For example, Jason Cowley used both the phrases old boy network and old school tie in Power, glory and the advance of the old school tie, published in The Times (London, England) of Thursday 24^th April 1997 [No. 65,872, page 21, column 1]:

The English ruling class is nothing if not adaptable. New Labour ¹ may be promising one of the greatest constitutional upheavals this century, but in reality it is unlikely that Tony Blair ² will do more than readjust the knot of the old school tie that can still open doors to all the best jobs. For the notion of the old boy network—exclusive university, the right school, family connections—shows little sign of withering away.

¹ Collins English Dictionary defines the name New Labour as follows: “a rebranding of the British Labour Party and its policies undertaken by Tony Blair ² and his supporters in the run-up to the 1997 general election in the United Kingdom and maintained during the Labour Party’s period of government under Blair’s premiership. Never an official title, it denotes the more right-wing/social democratic trend in Labour thinking and policy intended to make the party electable after its electoral catastrophes of the 1980s.”
—Cf. also the British-English adjective Tory-lite.
² The British Labour statesman Tony Blair (Anthony Charles Lynton Blair – born 1953) served as Prime Minister of the United Kingdom from 1997 to 2007.

The earliest occurrences that I have found of the phrase old boy network (also old boys’ network, old boy net, old boys’ net) are as follows, in chronological order:

1-: From The British in Germany: An Excess of Bureaucracy, a correspondence from Düsseldorf, in then West Germany, published in The Manchester Guardian (Manchester, Lancashire, England) of Friday 3^rd March 1950 [No. 32,254, page 6, column 6]:

Another reason for the tardy decrease in Control Commission personnel is the marked disinclination of authorities to get rid of the useless. Here bureaucracy is backed by the survival of the “old-boy-net” principle, which applied—but then constructively—during the war. Generally it is the junior officer, who has no “pull” with his sectional boss, who is “declared redundant.” The safest are those who work closest to their immediate superiors. Such people have an almost endless possibility of reprieve and there have recently been some flagrant cases of highly unsuitable members of the Control Commission being retained and even reinstalled in responsible and lucrative jobs.

2-: From a letter to the Editor, from Germany, in which a person signing themself ‘C. C. G.’ reacted to the above-quoted correspondence from Düsseldorf—letter published in The Manchester Guardian (Manchester, Lancashire, England) of Tuesday 4^th April 1950 [No. 32,281, page 6, column 7]:

It is difficult, if not impossible, to justify the continued existence of an Administration staff that accounts for nearly half of the entire personnel of the commission.
Apart from the “old boy net,” a main cause may be found in the fact that the Administration staff naturally spend a great deal of their time making, reading, interpreting, and altering an increasing stream of administrative instructions. Practically nothing is left to germinate for more than a few weeks before some new instruction is brought out. Result, more “empire building” by Admin.!

3-: From “Blat” Vital To Life In Land Of Russians, by Sefton Delmer, from the London Express Service, published in The Hamilton Spectator (Hamilton, Ontario, Canada) of Friday 27^th October 1950 [Vol. 105, No. 250, page 26, column 4]:

Here [i.e., in the Soviet Union] the Socialist plan has been realized. The means of production and distribution and exchange have passed under the control of the State, and that means the old-boy network of bureaucrats and party priviligentsia ³.

³ Used of the Soviet Union and other communist countries, the noun priviligentsia designated a class of approved intellectuals and Party bureaucrats enjoying certain social and economic privileges over ordinary citizens. [Incidentally, this occurrence of priviligentsia from The Hamilton Spectator of 27^th October 1950 predates the earliest occurrence of this noun that the Oxford English Dictionary (online edition, July 2023) has recorded—which is from the Syracuse Herald-Journal (Syracuse, New York, USA) of 4^th September 1953.]

4-: From Churchill ⁴ story—3: Young man in a hurry, by Colin Frame, published in the Evening Express (Liverpool, Lancashire, England) of Wednesday 1^st November 1950 [No. 24,680, page 4, column 3]:

SECOND-LIEUT. Winston Churchill, not quite 24 […]—fought for two months to join Kitchener’s Sudan Force […].
[…]
It was Kitchener himself who put up a stern defence against Churchill’s wish to take part in the campaign.
Even to the Prime Minister, Lord Salisbury, who interceded on the young man’s behalf, Kitchener said “No.”
He was clearly not—although he did not say so—prepared to have serving within a continent of him an uppish and youthful subaltern who dared to criticise his superiors in the Press.
But by one of those War Office loopholes known in the last war as “the old boy net,” Churchill succeeded in joining the Lancers, provided that he travelled at his own expense and no charge fell on Army funds if he was wounded or killed.

⁴ Winston Leonard Spencer Churchill (1874-1965) was a British statesman.

5-: From New “Unit Companies” In the Ruhr, a correspondence from Düsseldorf, by Don Cook, published in The Montreal Star (Montreal, Quebec, Canada) of Monday 5^th November 1951 [Vol. 83, No. 260, page 11, column 2]:

Now, however, comes the question, “What does deconcentration of heavy industry in the Ruhr actually mean?”
This question, asked of Ruhr owners and managers, usually provokes a tired smile. The British, when the question is put to them, have a handy colloquial response: “The old boy net it [sic] working just the same as ever.” The “old boy net” is a system with which the British are well acquainted for getting such things as price agreements or marketing arrangements accomplished between coffee and brandy instead of at formal board of directors meetings.

6-: From a correspondence from Blacon Camp, Chester, about the Army Emergency Reserve, published in the Manchester Guardian (Manchester, Lancashire, England) of Monday 23^rd March 1953 [No. 33,201, page 2, column 1]:

The Commanding Officer of an E.R. regiment is appointed by the War Office to raise a cadre of volunteers […]. The C.O.’s initial recruiting of the volunteers is a genteel form of press-ganging, carried out on what the Army calls the Old Boy net—“That you, Joe? Bill here, old boy.” […]
[…]
[…] Units generally have little difficulty in raising officer volunteers […]. The main shortage is of sergeants and corporals; once the Old Boy net has been exhausted, the unit is dependent on its National Service men for its N.C.O.s, and few of these have had time to rise much above the rank of lance-corporal during their two years in the Army.

7-: From Symptoms of Change in Argentina, by J. Halcro Ferguson, published in The Observer (London, England) of Sunday 12^th April 1953 [No. 8,445, page 7, column 1]:

It appears that Perón ⁵ alone may not be so strong as his sycophants and his own propaganda had led him to believe. It is also possible that he underestimated public knowledge and dislike of widespread corruption in high places. Argentina has traditionally been run by cuña, which may roughly be translated as the “old boy network,” rather than graft, and there was increasing resentment at Peronista leaders’ luxurious living at a time when everyone else was feeling the pinch.

⁵ Juan Domingo Perón (1895-1974), dictator of Argentina from 1946 to 1955.

8-: From the column Despatch Man’s Diary, published in the Evening Despatch (Birmingham, Warwickshire, England) of Tuesday 8^th December 1953 [No. 19,454, page 4, column 3]:

SHADES of the Western Brothers! The Old School Tie business is moving in on industry, you cads! Soon 100,000 wage-earning types will be knotting their “gents” natty neckwear—so that each will know another member of the Old Firm when they meet in the far-flung corners of the Commonwealth.
The firm to introduce its own Old Boy network is the mammoth Imperial Chemical Industries.
This month it is running a competition to decide on the pattern of the tie—blue, green or maroon background? With wavy stripes, or with both?
When a popular poll has decided, the ties will be made in I.C.I.-produced Terylene.
“It is expected to cost very much less than an all-silk tie, and will be unsurpassed for quality, durability and value,” says the I.C.I. magazine.

9-: From Are the Nazis coming back?, by Bruce Rothwell, published in the News Chronicle (London, England) of Monday 9^th August 1954 [No. 33,749, page 4, column 6]:

Nazism, let’s face it, is dead. […]
[…] And yet Nazism’s ugly cousin, an extreme form of arrogant nationalism, is very much in evidence in Germany just now. There is no need to re-create the Nazi movement when some of its staunchest supporters in the past—men who have given no evidence since of any real change of heart—are allowed to penetrate respectable democratic parties and become Cabinet Ministers, confidential advisers, foreign diplomats and leading administrators in the new West Germany.
[…]
Some are the “plants” of nationalist, ambitious Cabinet Ministers and M.P.s. Herr Oberlander ⁶ has been particularly active in providing jobs for his boys.
Others, in through the back door on the “old boy network,” are simply more efficient than men who lost their abilities in concentration camps.

⁶ A former Nazi official, Theodor Oberländer (1905-1998) was the then Federal Minister for Displaced Persons, Refugees and Victims of War in West Germany.