Since the First World War, in British English, the noun bowler hat, or bowler, has been used figuratively:
– as a symbol of civilian life as opposed to service in the armed forces,
– and, especially, in the phrases to be given one’s bowler (hat) and to get a bowler (hat), as a symbol of demobilisation or dismissal from the army.

The Welsh author Amy Roberta ‘Berta’ Ruck (1878-1978) evoked this symbol of return to civilian life in When Cupid Takes Off Khaki: The War Bride’s Future, published in the Sunday Pictorial (London, England) of Sunday 2^nd July 1916:

They are madness, these war weddings in haste that lead to repentance at leisure—in peace-time. For when Peace is really signed, and when our troops return at last, what about the girl who married the uniform? That uniform, drab-coloured but glorious, must be packed away. The dull bowler hat, the commonplace civilian clothes must take its place. Exit the “hero.” Re-enter the ordinary young man—the City clerk, the milkman, or whatever else he may be, of whom the girl said that she “might never have thought of marrying him.” But she’s married now. She can’t get out of that. Married to a man of whom she knows next to nothing. What awaits her?

The figurative use of bowler (hat) is first recorded in Soldier and Sailor Words and Phrases (London, 1925), by Edward Fraser (floruit 1925) and John Gibbons (1882-1949):

Bowler, to be given one’s: to be demobilised and returned to civil life.

The following definitions are from Songs and Slang of the British Soldier: 1914-1918 (3^rd edition – London, 1931), by the Anglo-Irish journalist and novelist John Brophy (1899-1965) and the New-Zealand born lexicographer Eric Honeywood Partridge (1894-1979):

Bowler, to be given one’s, to be demobilized. Bowler Hat, to get a, to be sent home, to be ‘sacked’. Officers’ slang; applied only to major and higher officers.

This is from the column Collie Knox Calling, published in the Edinburgh Evening News (Edinburgh, Midlothian, Scotland) of Saturday 28^th March 1942:

I do not intend to elaborate just one example, but unless work—and suitable work—is found for every over-age man given his bowler hat by the Army, there is going to be great trouble.

The figurative use of bowler (hat) gave rise to the noun bowler-hatting, denoting demobilisation or dismissal from the armed forces—as in the following from the column Echoes From Town, in the Nottingham Evening Post (Nottingham, Nottinghamshire, England) of Thursday 22^nd April 1943:

As Euclid Says.
Those whose schooldays included a course in the nowadays somewhat flyblown ratiocination of Euclid will recall how occasionally that succinct reasoner observed “Which is absurd.” That remark seems to be appropriate to certain manifestations by our military pundits concerned with man-power. Two cases within the writer’s personal knowledge prompt this criticism.
One friend of mine, aged 41, but registered at the age of 38, has just been called up. He is Grade 2, has had two bad attacks of pneumonia within three years, was doing useful newspaper work, and is obviously not cut out to make a vigorous Commando trooper. Yet another, a hale and hearty journalist in the early fifties, A1, is heart and soul a gunner. He served with great distinction in the last war with the guns, winding up as a temporary brigadier, and he is in command of important shore defences now. Yet he has been threatened with bowler-hatting. Now which is the more useful military asset—an unfit and untrained potential infantry private or an experienced, expert, and fit artillery specialist? Our War Office sometimes gives one a pain in the neck.

This figurative use also gave rise to the adjective bowler-hatted, meaning demobilised or dismissed from the army. For example, this is from the Somerset County Herald and Taunton Courier (Taunton, Somerset, England) of Saturday 18^th July 1959:

May Help ‘Axed’ Officers

After hearing an appeal by Sir William Grant, of Bristol, on Tuesday, Yeovil Rotary Club may form a subcommittee to advise ‘bowler-hatted’ Army officers looking for work in the area. Sir William is vice-chairman of the South-Western Regional Board for Industry.

In The Poor Gentleman (London, 1928), published under the pseudonym of Ian Hay, the British novelist John Hay Beith (1876-1952) made the narrator, Captain Shere, use the phrase to put the bowler hat on in the generic sense of to thwart—he and Corrie Lyndon are travelling by car in Normandy:

“Captain Shere, we are passing through a village now: the sign-board says ‘Pont-des-Briques.’ Have you any line of information about that?”
“Yes. Napoleon stayed here for nearly two years, with the Grand Army camped all round him, waiting to invade England. He had got a bit closer than the Kaiser, you see. He even brought Josephine and the Court along. But Nelson and Trafalgar put the bowler hat on that scheme.”