Dictionary coverage of 'on the line'
Christine Ammer, The American Heritage Dictionary of Idioms (1995), connects "on the line" with "lay on the line," which has yet another variant form, "lay it on the line." Here is Ammer's discussion:
lay on the line 1. Make ready for payment, as in They laid hundreds of thousands of dollars on the line to develop new the new software. [c. 1900] 2. lay it on the line. Speak frankly and firmly , making something clear. The professor laid it on the line: either hand in the term paper or fail the course. [c. 1920]
But Christine Ammer, The Facts on File Dictionary of Clichés, second edition (2006), has a slightly different take on the phrase "lay it on the line":
lay it on the line, to To speak frankly. This Americanism of the early twentieth century originally meant to hand over money (from about the 1920s). However, by mid-century it meant to speak plainly or categorically, and in the 1960s acquired still another sense, to lay something on the line, meaning to put that thing at risk (as in, "The Marines laid their lives on the line").
J.E. Lighter, Random House Historical Dictionary of American Slang (1997) has three definitions for phrases that include the phrase "on the line":
lay {or put} it on the line {fr the gambling sense 'to put money on the line {space on a craps table for placing chips}'} 1. to put down or hand over money. Now colloq. or S[tandard] E[nglish]. [First cited occurrence:] 1929 D[amon] Runyan, in OED2: My rent is way overdue...and I have a hard-hearted landlady....She says she will give me the wind if I do not lay something on the line at once. ...
2. to explain; speak one's mind firmly and directly. Now colloq. [First cited occurrence:] 1944 Butler & Cavett Going My Way (film): Like Tony says, I'm gonna lay it on the line.
on the line engaged in prostitution, esp. in a red light district. [First cited occurrence:] 1910 in Roe Prodigal Daughter 83: Oh if everybody did know the awful shame and degradation of such a life there would surely be very few girls "on the line."
'Lay it on the line' as 'bet or spend money on it'
The earliest Google Books match for "lay/put it on the line" in the sense of "bet on it" is from "This 'Mere Mechanical Toy' Was Fifty Feet Long!" in Popular Science (December 1931):
...how in the name of common sense could the monster be running round with a clothed young lady in its mouth? It just doesn't stand to reason unless I'm getting cock-eyed in my old age. And anyway, if by some freak of nature a girl had strayed into the dinosaur age, it's a perfectly safe bet of a million to one that she'd have been clothed in just exactly nothing or quaintly wrapped in a rapidly decomposing piece of hide. Am I right? You can put it on the line that I am.
And from E.A. Batchelor, "Gasoline-Buggy Beginnings," in The Rotarian (March 1934):
So that good old Anglo-Saxon custom of trying to tax anybody to death if he seems to be getting along was instituted. Most of the roads were privately controlled in those days; turnpikes upon which appeared at irritating intervals toll gates where the traveller "laid it on the line" until it hurt. The steam carriages were just made to order for the toll-gate boys. The steam carriages were just made to order for the toll-gate boys.
And from Raymond Chandler, The Big Sleep (1939):
"Does there have to be something else?"
"Yes."
She stared at me a little puzzled. "There is. The woman said there was a police jam connected with it and I'd better lay it [the blackmail money] on the line fast, or I'd be talking to my sister through a wire screen."
'Lay it on the line' as 'speak plainly'
The "spoke bluntly" sense of "lay it on the line" first appears in "Chi Feels News Segs To Hold Continuous Sock Audience Pull," in The Billboard (September 29, 1945):
Occasion was forum held by Chicago Radio Management Club. Webs don't figure on losing many sponsored programs, tho some paring of commentators was envisioned, to be replaced by on-the-spot boys—both local and national.
Here's the way the boys laid it on the line: [Quotations from four network news programmers omitted.]
From "What Businessmen Themselves Foresee," in Kiplinger's Personal Finance (July 1949):
Hundreds of letters from working businessmen have recently been received by the editor of this magazine. The men laid it on the line—freely, frankly, without any pose. They told how their own business was faring, and what they foresaw for the future. They were practical and down to earth. Here is a brief report on what they thought and said.
And from "Our Enemy Is Confident; and He Tells Us, 'This Is War'," in Life magazine (December 18, 1950):
Our Communist enemy laid it on the line with a sureness and confidence which went beyond the familiar tone and language of mere propaganda. Probably the most meaningful words of the week were spoken at Lake Success by Andrei Vishinsky, the Soviet Foreign Minister. He was speaking of a mild little resolution, put up by the U.S. and five other powers. It simply asked the Chinese Communists to withdraw their troops from Korea.
Conclusions
According to Random House Historical Dictionary of American Slang, both senses of "on the line" that the OP asks about derive from the same source: a reference to the physical line on a craps table where gamblers place their chips in making a bet. That dictionary finds examples of the two senses of the phrase from 1929 (in the sense of "to put down or hand over money") and 1944 (in the sense of "to speak one's mind one's mind firmly and directly"). Google Books searches turn up instances of the phrase in each sense starting in 1931 and 1945, respectively.
None of the other reference works I consulted offers any documentary evidence contradicting the RHHDAS's findings.
Best Answer
During a conventional war, keeping troops supplied with arms, food, and clothing has long been said to occur along a line of supply/supply line whose interruption is a primary military goal:
In agriculture or manufacturing, a similar logistical phenomenon may be called a chain of distribution, the metaphor usually extended by designating stations along this path as links:
First, America
In early 20th c. America, two writers choose a different metaphor, comparing the transportation of goods to water, oil, or gas moving through a pipeline:
Both the extension of the metaphor in the first example to the participle conduiting and the explicit introduction of the metaphor in the second suggest that here, pipeline is just that: metaphoric language, not common usage.
Beyond these two outliers, in the daily press on either side of the Atlantic (or Pacific), the notion of a supply line as pipeline does not appear until 1942, after America’s entry into WWII, supply line remaining by far the more frequent term.
The quotation marks bracketing pipeline in this syndicated column suggest either military jargon, a relative new usage, or both. If it had been a common term among the American military in the First World War, however, one would expect more than a single instance among American dailies.
US Army Quartermaster Corps
Since 1775, the American military branch responsible for determining how many pairs of shoes a soldier requires under what conditions and delivering them when needed has been the Quartermaster Corps. Starting from vitually nothing when the United States entered the First World War, by Armistice Day, the corps had been equipping some 10,000 soldiers a day and shipping them to Europe. A 1946 monograph acknowledges the advantages of the pipeline concept and its general use in the corps:
While Bradford considers the ultimate origin of the term “obscure,” [17, note 86] he also points out that supply line as pipeline was a key concept in a study published four months before the end of the First World War, also mentioned in a Dec. 1918 history of the Requirements Division of the corps as having been authored by a “Major Griffin”:
The method of calculation and the more granular statistics required by the pipeline model proved too complicated for practical use at the time, but its best features “resulted in the adaptation of the ‘Pipe Line’ idea of requirements with which every one interested in Requirements is now familiar” [Requirements, 31].
It is thus not too far a stretch to suggest that 1) the Quartermaster Corps preserved the pipeline metaphor while discarding certain aspects of Griffin’s model and adapting others, 2) that the expression remained jargon and was still in use — or quickly dusted off — when the United States entered the Second World War, and 3) that as American and Commonwealth forces began to coordinate logistics, the latter took up the pipeline metaphor and occasionally used it themselves.
Britain, Australia, and Back to America
Some six months before the Texas paper, a British daily uses pipeline to denote a military supply line by employing a simile:
This is followed by several other uses in the same year:
And also in a 1942 Australian paper:
Throughout the war years, there is scattered use of the metaphor:
And in Australia, though the second example cites a British officer, the first is Prime Minister John Curtin:
And back to America:
In the course of the war and at its end, not just supplies, but also personnel could be “in the pipeline.” The expression is expanding from a locative expression to more of a temporal one: whatever is in the pipeline is coming, but not yet. One can imagine, for instance, the desert soldiers envisioned in the Breckenridge TX article wondering when their new shoes would arrive and hearing from the quartermaster, “They’re in the pipeline!”
Post-War Years
After the war, the expression transitions from military usage such that virtually anything requiring a process to complete could be said to be “in the pipeline.”
Twenty years before, it would have seemed odd suggesting that an airport could be “in the pipeline”: the vehicle of the metaphor, if there still is one, can be expanded from moving something from A to B to how long it takes to get here. The same transition occurs in British English:
Conclusion
In the same year, 1942, only months after Pearl Harbor, pipeline as supply line appears in British, American, and Australian newspapers. The two isolated American uses in 1916 and 1922 can merely suggest American provenance.
One is on much firmer ground, however, when the 1918 use among the US Army Quartermaster Corps enters into the mix: the pipeline metaphor, if not the exact model constructed by Maj. Griffin, appears in an identical sense twenty-six years later. The virtually simultaneous attestations in British and Australian newspapers suggest that military logistics specialists from those countries found the metaphor as apt as Bradford did and began to use it themselves. From that jargoned usage, it “escaped” into the daily press, eventually attaining common usage in every flavor of English in the idiom in the pipeline.
In those early attestations, the metaphor was more place than process: today, it can be both. A heroin pipeline is a smuggling route; in the pharmaceutical industry, a drug pipeline is the aggregate of all compounds at various stages of development within a particular company.
When money is concerned there is still some sense of the older usage. After all, one speaks of cash flow or liquid assets, so it makes sense that money can still move from A to B through a metaphorical pipeline:
If there were any American who knew about military supply pipelines, it would have been former Gen. Eisenhower, who served as Supreme Commander of the Allied Expeditionary Forces in Europe.
Otherwise, today across headlines in the Anglosphere, in the pipeline is all about plans and process: