This is just a placeholder for an answer since I have to deal with some emergency plumbing issues at the moment.
EDIT: I an becoming more convinced that the origin of the phrase "cut|take a voluntary" is ultimately military. The earliest unambiguous military attestation I'd been able to find was in the context of conscripts in the US Civil War in a book written by a Brit who was visiting the States at the time (see below). But @TinfoilHat has floated in a comment below an attestation from a decade earlier, in a steeplechase context. In further support of the military origin, here is a description of a veritable "epidemic" of self-maiming in the British army from ON
THE ENLISTING, THE DISCHARGING,
AND
THE PENSIONING OF SOLDIERS by Henry Marshall (Edinburgh, 1839)
The practice of maiming has been so frequent in some regiments as to appear epidemic. The author belonged to a regiment in which nine men maimed themselves in the course of six weeks and in every instance the injury was presumed to have been inflicted voluntarily. The explosion of their own muskets was the plan they invariably adopted to effect their purpose and the injury was always in a foot or a hand. They all, as may be inferred, attributed the explosion of the musket to accident and not to design, but one man who maimed himself in a necessary found much difficulty to account for his taking a loaded musket into a place of that kind. Very lately when Regiment was at Cork and about to embark for the West Indies, four of the men made their appearance with the first joint of the thumb of the right hand mutilated. Voluntary mutilation sometimes takes place under very extraordinary circumstances even during a conflict with an enemy.
From Border and bastille by George A. Lawrence (a Brit), which was published in New York in 1863 and is catalogued by the US Library of Congress under United States--History--Civil War, 1861-1865:
On the outbreak of the war, volunteers enlisted
in the Federal cavalry, who --far from being able
to manage a horse-- could not bridle one without
assistance; and a conscript, who could keep his
saddle through an entire day, without "taking a
voluntary", was considered by his fellows as a credit
to the regiment, and almost an accomplished
dragoon. Such a thing as a military riding-
school has, I believe, never been thought of,
away from West Point; the drill is simply that of
mounted infantry. Things are better now than
they were; a Federal cavalryman can at least sit
saddle-fast, to receive and return a sabre-cut; there
have been some sharp skirmishes of late, and,
allowing for exaggeration, Averill s encounter
with Fitzhugh Lee brought out real work on
both sides.
By putting the phrase "taking a voluntary" in quotes, Lawrence seems to be quoting a phrase he has heard while visiting the encampments of the US Army, one not already familiar to a British audience.
In the chapter where the paragraph above appears, he is far more concerned about the way horses are treated by the northern Army than about the plight of the men. He speaks of the squalor of encampments and the lack of esprit du corps among the lower ranks of soldiers, the result of which is a high rate of mortality for the horses:
That utter absence of
esprit du corps and soldierly self-respect, has cost the Federal treasury many millions; nor will the drain ever cease till "re-mounts"
shall be no more needed.
The foregoing remarks apply exclusively to
the tenue of the privates and non-commissioned officers; those of
superior rank that I met were tolerably correct, both in dress and
equipment; several, indeed, were mounted on really powerful chargers,
and rode them not amiss, though with a seat as unprofessional as can
be conceived.
P.S. Evidence of the phrase origin would be an attestation (ideally additional attestations) for the posited explanation. We know the phrase is used in hunting; slang dictionaries before Green's have glossed the phrase as "to fall off a horse while hunting" and attestations can be found. But the earliest attestation so far is a cavalry context. Did the phrase begin in hunting or cavalry? And if the latter, was it a long-standing British cavalry phrase predating 1863 or was it coined as American military jargon or American military slang? In the same chapter, George A. Lawrence does indeed refer to US military jargon ("re-mounts"). So it is not implausible that he was citing two terms he had heard used in the US Army on his visit to the States. The question is open -- from a lexicographical perspective.
P.P.S. You might be surprised by how often you'd hear the word "bespoke" these days here in the good old US of A. The borrowing is a two-way street.