W.S. Farmer & J.L. Henley, Slang and Its Analogues, vol. 2 (1891), says that "dead to rights" means "certain; without doubt," and asserts that it is simply an amplification of the earlier term "to rights," meaning "completely to one's satisfaction." Dead appears in a similarly amplifying way in such current phrases as "dead broke," "dead certainty," "dead heat," and "dead ringer."
Christine Ammer, The American Heritage Dictionary of Idioms (1997) gives a sense of how much the meaning of the phrase has evolved:
dead to rights In the act of committing an error or crime, red-handed. For example, They caught the burglars dead to rights with the Oriental rugs. This phrase uses to rights in the sense of "at once."
Harold Wentworth & Stuart Flexner, Dictionary of American Slang, first edition (1960) gives both meanings:
dead to rights 1 Certain. c1880. Colloq. 1939: "You've got him dead to rights...' Gardner, D. A. Draws, 185. 2 Caught in the act of or irrefutably accused of an illegal, immoral, unethical, or antisocial act.
But the Robert Chapman & Barbara Kipfer, Dictionary of American Slang, third edition (1995) agrees with Ammer's narrower definition:
dead (or bang) to rights adv phr by 1859 With no possibility of escape or evasion; in flagrante delicto; red-handed: ...was caught "dead to rights" and now languishes in the city Bastille—San Francisco City Argus
UPDATE (9/17/2013): I did some checking in Google Books search results and found a source that probably explains Chapman & Kipfer's "by 1859" dating. George Matsell, Vocabulum; or, The Rogue's Lexicon (1859), contains this entry:
DEAD TO RIGHTS. Positively guilty, and no way of getting clear.
To like effect is a glossary entry in George Burnham, Memoirs of the United States Secret Service (1872):
DEAD TO RIGHTS, caught, with positive proof of guilt.
So it appears that Farmer & Henley didn't fully grasp the sense of "dead to rights" as the phrase was used in the latter half of the nineteenth century, though the meaning that Slang and Its Analogues ascribes to the term certainly does appear in some contemporaneous writings. For example, from "Born and Raised in Mobile," in Sweet & Knox, Three Dozen Good Stories from Texas Siftings (1887):
"You never heard what became of Wes Brutus?" inquired the driver.
"No; I expect he's sloshing around, though. Wes was a bully boy."
"Well, now, pardner, your [sic] just dead to rights; he was for a fact—ah! you folks are going to try Foot & Walker's line, are you? That's right; that's correct.
FURTHER UPDATE (1/24/2017): The earliest match for "dead to rights" that an Elephind search of multiple old newspaper databases returns is from "Pickpockets at the Depot: Caught in the Act—One of the Gang in Prison," in the [Harrisburg] Pennsylvania Daily Telegraph (August 14, 1862):
This morning, upon the arrival of one of the trains on the Northern Central Railroad, a passenger named Loring Gillis, of Buffalo, New York, felt a hand inserted in one of his pockets, and before it could be removed, he turned suddenly around, and succeeded in coloring [sic] its owner, a dapper, flashy dressed individual of thirty or thereabouts, with "pickpocket" written all over him. Upon feeling for his pocket-book, containing about two hundred dollars, Mr. Gillis found it was gone. The "chevalier," like all his class when caught "dead to rights," tried to shut Mr. Gillis' eye up with the "virtuous indignation" game, but finding that no "go," tried to close it with one of his fists. Here, too, it was foiled, for Mr. Gillis showed a superior proficiency in the "manly art," and the "chevalier" was finally forced to succumb. During the scuffle two individuals, supposed to be pals of the pickpocket, were seen to run from the crowd, but from some cause were not pursued and arrested.
Best Answer
The earliest use of jam-packed I have been able to find occurs in a review of a children’s program — the finale was an “emblematic representation of the majesty of Britannia” — in late 19th c. Cardiff:
When an author wishes to indicate a new coinage, the usual choice is simply to bracket it with quotation marks, especially if its meaning is obvious. A long appositive such as this one suggests jam-packed is still warm from the oven.
The first attestation in an American source occurs in a short story syndicated to a number of newspapers — not by an American, but by the popular Scottish author Samuel Rutherford Crocket:
Written in the Lowland Scots eye dialect for which Crocket was noted, the story would be the last place one would look for an Americanism. The story also appeared in February 1896 in the Pocket Magazine and finally in a collection of Crocket’s short stories, Love Idylls, published in London in 1901. It is perhaps this publication that gave the Online Etymological Dictionary or its source a 1901 date for the first appearance of the word.
Whatever the case, the Scotsman Crocket and the anonymous Cardiff “Observer” prohibit otherwise straight linguistic lines from North America to the United Kingdom which the sightings of jam-packed in American sources after the turn of the century would suggest. After these two uses, British sources seem to go silent until the end of the 1910s.
In the year Crocket’s story collection went to press, however, a bit of filler appeared in at least two American publications:
While jam-packed doesn’t become especially frequent in the US until the mid-1920s and later, its use is spreading across the country:
If the word appears in an official publication of the Evangelical Lutheran Augustana Synod of North America, overwhelmingly composed of Swedish immigrants and their descendants in the Midwest, then jam-packed has truly arrived in American English.
What stands out in these examples is that except for the filler about advertising and the ad itself, jam-packed appears in works of fiction imitating, as in Crocket, everyday, informal speech. The speaker in the church publication is merely a slangy straw man invented to be set to rights by an older Swedish Lutheran that social events are not the main mission of the church.
The first appearance in an Australian newspaper, at least among those digitized by the National Library of Australia, occurs in a Melbourne paper:
And in 1919 Northern Ireland, a headline proclaims
Across the United Kingdom, in the 1940s, jam-packed has 37 hits in the British Newspaper Archives (BNA) especially in articles about American films, most likely echoing Hollywood publicity. Each decade until 1980 has over a hundred hits, the 80s only 69, but 1,167 in the 1990s. Despite Crocket, there are no other Scottish sources from his time and thereafter that would suggest that jam-packed was particularly common in Scotland.
So...British or American?
Your question concerns the first attestation of jam-packed. That was answered by the first few paragraphs of my answer: Cardiff and a Scottish author publishing in American newspapers. Given the limits of what from this period is digitized and what is not, and that beyond its use in advertising, jam-packed was a slang expression not likely to appear often in print, evidence for a clear line across the Atlantic, however, is at best ambiguous.
While it is hardly difficult to draw linguistic lines from Scotland to Northern Ireland, one might expect the BNA to contain a few Scottish newspapers that use the expression at the turn of the century. Perhaps someone with access to the Scottich National Library digital holdings might have better luck.
It is, however, difficult to imagine that Crocket, despite his popularity in America, is the source of the word in North America. Jammed and packed were a common collocation:
Jam followed by a participial phrase headed by packed was also not uncommon:
In other words, it is not outside the realm of possibility that jam-packed arose independently on both sides of the Atlantic without initial influence one way or the other. More pre-1900 attestations might add clarity, but these, if they exist, are not currently available online.
In moderate to rapid informal speech, to avoid the “reset” necessary to pronounce the following p, the d following the collocation jammed packed would likely be elided, as in red and purple. The result would be jam packed, which may ultimately be the independent source for both Crocket and the American usage. All jam, packed would require is a bit of reparsing and dropping the rhyming reduplication.