In British English, Jack Jones (rhyming slang for alone, or for own in on one’s own):
– is occasionally used as an adjective or adverb meaning alone;
– is chiefly used in the phrase on one’s Jack Jones, meaning on one’s own.

Rhyming slang is a type of slang that replaces words with rhyming words or phrases, typically with the rhyming element omitted. For example, in the phrase not to have a scooby, meaning not to have a clue, scooby, rhyming slang for clue, is short for Scooby Doo, the name of a cartoon dog which features in several U.S. television series and films.

In Jack Jones, there is no truncation, contrary to the usual rhyming-slang formation. The following explanations, by the British lexicographer Jonathon Green (born 1948), are from Cockney, published in the Oxford English Dictionary blog on Friday 17^th August 2012:

The original rhyming slang, which was a conscious attempt to mystify the uninitiated, depended on the omission of the rhyming element, for example: ‘Barnet fair’/‘hair’ (1857) to barnet (1931); ‘china plate’/‘mate’ (1880) to china (1925); ‘Hampstead Heath’/‘teeth’ (1887) to Hampsteads (1932); and ‘Sweeney Todd’/‘flying squad’ (1938) to Sweeney (1967). However this was by no means a rule, and there exist a number of terms in which the entire compound is pronounced—hence Adam-and-Eve/‘believe’ (1925), cocoa/‘say so’ (1936), or tea-leaf/‘thief’ (1903).

—However, Jack Jones is truncated to Jack in quotation 16, below.

The earliest occurrences of Jack Jones that I have found are as follows, in chronological order:

1-: From Soldier and Sailor Words and Phrases (London: George Routledge and Sons, Ltd., 1925), by Edward Fraser and John Gibbons [page 130]:

JACK JONES: Alone, e.g., “I am going out Jack Jones to-night”.

2-: From the column Questions We Ask, published in the Weekly Dispatch (London, England) of Sunday 14^th February 1926 [No. 6,487, page 10, column 4]:

What is the exact meaning of the phrase “getting down to brass tacks” which has been used by Lord Weir on the housing question?
The people of the United States have a corresponding phrase, “getting down to cases,” which is probably derived from the courts, where decisions are given on analogies with reported cases.
Sailors have been known to explain it by reference to nautical tacks, shoemakers by reference to the fact that brass tacks are the hardest wearing. Soldiers claim that it is really only rhyming slang, brass tacks standing for hard facts.
The purists would probably object that facts does not rhyme at all with tacks. But what of the other rhyme “I am here on my Jack Jones” for “I am here on my own, or alone”?

3-: From a letter presented as evidence during a trial at Croydon County Police Court, published in The Streatham News (London, England) of Friday 11^th December 1931 [No. 2,190, page 9, column 3]:

“Write and let me know when you are coming up to town. I am on my Jack Jones once more.”

4-: From the account of a police-court case, published in The West London Press Chelsea News (London, England) of Friday 28^th October 1932 [No. 3,760, page 3, column 4]:

Mr. Watson: Have you any witnesses to call?—Prisoner: None at all, sir. I was all on my “Jack Jones.” (Laughter).

5-: From A Clergyman’s Daughter (London: Victor Gollancz Ltd, 1935), by the British novelist and essayist George Orwell (Eric Arthur Blair – 1903-1950) [page 156]:

‘Oh, well! Time Doris and me was moving. Got a ’portant business engagement in Leicester Square.’ Here she nudged the other girl with her hip, and both of them sniggered in a silly mirthless manner. ‘But, say!’ added the taller girl confidentially, ‘ain’t it a bloody treat to ’ave a good night’s kip all alone once in a way? Wish I could. All on your Jack Jones with no bloody great man’s feet shoving you about. ’S all right when you can afford it, eh?’
‘Yes,’ said Dorothy, feeling that this answer was expected of her, and with only a very vague notion of what the other was talking about.

6-: From an account of the 16^th annual County Conference of the British Legion in Berkshire, published in The Reading Standard (Reading, Berkshire, England) of Friday 11^th December 1936 [No. 3,341, page 21, column 3]:

Complaint was made as to the procedure adopted for the issue of tickets in connection with the Albert Hall Festival of Remembrance, several delegates stating that their branches had only one ticket. One delegate said no one wanted to go up to London for the festival on his “Jack Jones”—by himself.

7-: From Soldiers in Training. Advice and Hints for Young Soldiers, with Traditional Bugle Calls and Slang Vocabulary (London/New York: Warne & Co., 1939), by ‘Soldierman’—as quoted by ‘Reflex’ in Reflexions by Reflex, published in The Essex Newsman and Maldon Express (Chelmsford and Maldon, Essex, England) of Saturday 9^th December 1939 [No. 3,647, page 4, column 1]:

There are eight pages of “Soldiers’ slang.” [I] choose a few:
Barker.—Sausage.
Burgoo.—Oatmeal porridge.
Casabianca.—The last one.
Clink.—Guard-room or cells.
Coffin Nails.—Cigarettes.
Fiddler.—A bugler.
Floaters.—Dumplings.
Gaspirators.—Gas masks.
Gongs.—Medals.
Gunfire.—Early morning tea.
Jack Jones.—Alone.
Jake.—Honest and just.
Jankers .—Punishment.
Poodelfaker [sic].—Social climber.
Pop Wallah.—Teetotaller.
Pozzy.—Jam.
Pronto.—Quick!
Rosebuds.—Potatoes.
Sticks.—Drummer.
Sand.—Sugar.
Sugar.—Money.
Tommy.—Bread.
Waterloo Day.—Pay-day.
Zepp in a Cloud.—Sausage and mashed.

8-: From Peter Wilson’s Sport Column, published in the Sunday Pictorial (London, England) of Sunday 29^th September 1946 [No. 1,646, page 13, column 1]—the following is about speedway:

The Wembley v. West Ham show had a real “champagne launching” when Tommy Price beat Jack Parker in the Sunday Pictorial match race championship. […]
[…] The A.C.U. steward, A. G. Pickering, […] had to make a hair-trigger decision. When they came up to the tapes for the second heat Pickering flashed the green light to show that the race was on.
Price put up his hand and indicated that he had forgotten to switch on his [petrol?], but, by then, it was too late for him to delay the start, and Pickering awarded the heat to Jack Parker—but sensibly did not insist that he cover the four laps “on his Jack Jones.”

9-: From an account of a football match between Dundee and St. Mirren, published in The Sunday Post (Glasgow, Lanarkshire, Scotland) of Sunday 5^th March 1950 [No. 2,323, page 22, column 4]—here, the use of the phrase on one’s Jack Jones may have been induced by the fact that the footballer’s name was Jack Jones:

In the opening stages, the play of Jack Jones at centre must have made the Love Street officials very happy. His forward slips and intelligent distribution were a delight, even though they were wasted.
But, presented with easy scoring chances, Jack was a different boy. Twice, all on his Jack Jones, he got the ball all mixed up with his legs.

10-: From I joined the long line of homely people, about “the little people of Britain” who paid homage to the coffin of King George VI lying in state at Westminster Hall, London, by Victor Thompson, published in the Daily Herald (London, England) of Wednesday 13^th February 1952 [No. 11,208, page 4, column 5]:

I spoke to the man beside me, 64-year-old Bill Fisher […], and he said:—
“It’s like this, see. My wife’s gone to stay with our married daughter down in Sussex, and I’m on my Jack Jones like, so I thought I’d come here and see what’s what and have something to tell her when she comes back.”

11-: From the News Chronicle (London, England) of Wednesday 25^th November 1953 [No. 33,532, page 5, column 7]:

COR—ON HIS JACK JONES!
By ROYER DAVIS

“COR,” said the street corner Cockney watching the fair-haired young man who was so energetically knocking at dingy front doors, “—He’s on his Jack Jones.”
The 28-year-old black-coated figure so described as being unaccompanied (the phrase is rhyming slang for “on his own”) was John Eden, the Tory candidate, whom this onlooker, experienced in elections, had expected to see with the usual entourage.
[…]
John learned debate by addressing shipbuilding workers in Southampton—on his Jack Jones, naturally.

12-: From Scouts Log, by ‘Black Panther’, published in the Midland Counties Tribune (Nuneaton, Warwickshire, England) of Friday 16^th March 1956 [No. 2,089, page 2, column 3]:

1^ST NUNEATON GROUP
[…]
We have a lone camper too, J.B. spent a week-end camp on his Jack Jones. The scare was on at 9-30 p.m. Sunday night for the Lone Wolf had not returned and the witching hour was very near before he eventually turned up. Comment… It would have taken Big Ben, in the room to have roused him the following morning.

13-: From Paul Temple, by the British author Francis Durbridge (1912-1998), published in the Edinburgh Evening News (Edinburgh, Midlothian, Scotland) of Thursday 18^th July 1957 [No. 26,297, page 10, column 1]:

LOOKS LIKE YOU’LL HAVE TO HANDLE THE BLOCK CAR ON YOUR JACK JONES, FRED, BUT YOU KNOW THE DRILL
LIKE THE BACK OF MY HAND… I’LL BE ALL RIGHT, HARRY

14-: From the caption to a photograph of Humphrey, a dog fond of travelling by bus, published in the Evening News (Portsmouth, Hampshire, England) of Friday 13^th September 1957 [No. 25,004, page 7, column 1]:

“This one’s of me on the bus, travelling on my Jack Jones, the way I liked it.”

15-: From End to Come, a short story by Paul Feakes, published in the Lincolnshire Echo (Lincoln, Lincolnshire, England) of Monday 30^th September 1957 [No. 19,062, page 4, column 4]:

Now this first member of the criminal classes, say me, for instance, is standing in a saloon bar one evening having a quiet drink on his Jack Jones.

16-: From A Wayfarer’s Notebook, compiled by ‘Puck’, published in the Huddersfield Weekly Examiner and West Riding Reporter (Huddersfield, Yorkshire, England) of Saturday 5^th September 1959 [No. 30,432, page 6, column 3]:

Rhyming Slang
WAYFARER’S paragraph about the Yorkshire Dialect Society in “The Weekly Examiner” of August 22, prompts me (Miss Dorothy Pearson, Hampstead, writes) to comment on the transformation which has overtaken the famous Cockney rhyming slang during my twenty years’ residence in London.
Many non-Londoners will be familiar with the expressions “Up the apples and pears” (up the stairs); “Would you Adam and Eve it?” (would you believe it) and so on. The interesting thing about these Cockney versions of standard English was that they were always much longer than the correct phrase.
It is perhaps not surprising, therefore, that in this streamlined age the Cockney versions of everyday expressions have gradually been contracted into what might aptly be described as a “speed merchandise.”
The old “tit-for-tat” (hat) has become merely a “titfer”; “plates of meat” (feet) are now only “plates.”
A Cockney Poser
THE non-Cockney has therefore to work backwards through a series of translations to arrive at the meaning of the present day Cockney phrase; or (more often) single substitute word.
This change has taken place gradually and those of us who live in London have, for the most part, kept pace with it.
But I must confess to having been a little disconcerted a few days ago when my Cockney waitress (who has served me lunch for the past eleven years) greeted me with “Hullo, Ducks, you on your Jack today?”
However, I recovered sufficiently to work out that this perversion of the Queen’s English was meant to imply “Are you on your Jack Jones?” rhyming with “only-ones”; in other words “Are you alone today?”
I wonder what a visitor fra t’North would have made of it?