The phrase home, James (and don’t spare the horses) is used as a humorous exhortation to a driver. James is a generic ‘posh’ Christian name arbitrarily given to a driver.

This phrase was originally used in the shorter form home, James.

I have found an early occurrence of this in the story of Chimmie Fadden, who explains to a friend of his why he is standing by a carriage and wearing a livery: Chimmie says that he saved from harm a lady who was “doin missioner work in der slums” of the Bowery, New York City, and that she showed her gratitude by employing him as her footman (i.e., as her carriage attendant); this story, first published in the New York Sun, was reprinted in several U.S. newspapers in 1892—for example, in The Atchison Daily Globe (Atchison, Kansas) of Thursday 13^th October:

“’Scuse me now, dats me loidy comin outer de shop. I opens de door of de carriage an she says, ‘Home, James.’ Den I jumps on de box an strings de driver.”

Another occurrence of the shorter form is from the column A Little Nonsense, published in The Washington Herald (Washington, D.C.) of Monday 22^nd June 1908:

So Romantic.

“Say!”
“Well?”
“It must be like living in the pages of a novel to be able to lean out of the carriage window and say ‘Home, James.’”

The earliest occurrence of the extended form that I have found is from the following article, published in The Inter Ocean (Chicago, Illinois) of Sunday 29^th January 1911:

HOBO KING STILL OUT ON 1894 STRIKE; GOES TO JAIL IN AUTO
Thomas Alexander Michael Ignatius McCormick Quit During Pullman Walk-Out—His Sympathy Has Not Yet Permitted Him to Work.

Special Dispatch to The Inter Ocean.
HAMMOND, Ind., Jan. 28.—Thomas Alexander Michael Ignatius McCormick loves his fellow men. He feels deeply for the oppressed and humbled working man. His sympathy was so profound that when the Pullman employees went on a strike in 1894 Thomas A. M. I. McCormick went on a strike, too. He has not been to work since, so great is his feeling over the matter. As king of the hoboes he was brought up before the judge this morning and sentenced to three months in the county jail.
Besides being of a deeply sympathetic nature, Thomas A. M. I. McCormick is as a shrinking violet when it comes to modesty. He does not seek to thrust his personality upon others. He has been known to wait quietly behind a water tank when he desired to journey from one place to another and creep quietly, avoiding the eyes of his fellow men, into the entrance of some side-door Pullman to keep from being spoken to by some seeker after truth or information.
Naturally, therefore, when Chief Albert T. Lewis suggested that Thomas A. M. I. McCormick make the journey by train to the institution where he is going to enjoy the colder weather, the modesty of the McCormicks asserted itself. He informed Judge William Riley that ever since the battle of the Boyne the blushing primrose had been as a brass band to the modesty of the house of McCormick. And he got away with it.
An automobile was secured to carry the king of hoboes in true regal fashion to the East Chicago sanatorium. Between two lines of bowing courtiers he descended from the court room to the waiting car and was assisted in by none but the chief of police himself.
As Thomas, etc., McCormick sank comfortably back on the leather cushions, he produced the last mortal remains of a cigar from its concealment in some part of his frayed costume and as he swept a match majestically along the spotless varnish of the automobile he nodded to the multitude which had assembled to witness his departure and remarked to the chauffeur:
“Home, James, and don’t spare the horse.”

The second-earliest occurrence of the extended form that I have found is from the review of The Deep Purple, a stage play presented by the Poll Players—review published in the Wilkes-Barre Times-Leader (Wilkes-Barre, Pennsylvania) of Tuesday 6^th August 1912:

In regard to the play itself it can only be said that it is particularly interesting at the present moment in view of the graft ¹ disclosures in the big city. The line “not in New York City” given in reply to a query if the officer believed anybody was honest, brought forth a round of applause last night. The workings of the badger game ² was disclosed in a manner that is bold yet no offense is given. The lines teem with strength and yet this in no small part is due to the slang of the criminal world. For instance “Frisco Kate” says “Spring Laylock or up the river you all go.” Following this statement in a tense scene when she realizes the new arrival from the West who has taken a grip on her heart has been betrayed she orders the detective whom she has cowed to call her a taxi “home James and don’t spare the horses.” Other examples are “I could kill you and get a pension for it.” “When a girl like you cries you cry with her.”

¹ Here, graft denotes bribery and other corrupt practices used to secure illicit advantages or gains in politics or business.
² The phrase badger game denotes an extortion scheme in which the victim is lured, usually by a woman, into a compromising situation and is then surprised and blackmailed by an accomplice.

E. S. Jordan (Sales Manager of The Thomas B. Jeffery Company, a U.S. automobile manufacturer in Kenosha, Wisconsin) used the phrase figuratively in an article published in The Boston Herald (Boston, Massachusetts) of Sunday 10^th January 1915:

WONDERFUL TIMES WILL SOON BE HERE
Country-Wide Prosperity and Big Business Will Be Had by Spring.
By E. S. JORDAN,
Thomas B. Jeffery Co.

Within the next 90 days $25,000,000,000 in new business will be staggering every line of industry in this country. In 90 days more this figure will have doubled.
[…]
Just as the manufacturer of what may be termed home necessities has jumped into harness, so will the automobile manufacturer of established reputation say to his superintendents, “Home, James, and don’t spare the horses.” Some of the western states already show an increase in the number of automobiles owned, jumping from one in every seventeen families to one in every ten. Massachusetts has one automobile to every half mile of improved roads—and practically every road in Massachusetts is improved. Automobile dealers are contracting for a greater number of cars than ever before, and all are basing their contracts on business already in sight.

An early variant of the phrase occurs in an item by W. L. Doyle, care agent, San Francisco, California, in Among Ourselves, published in The Santa Fe Magazine (Chicago, Illinois) of January 1921—The Santa Fe Magazine was “a monthly publication devoted to the interests of the 75,000 employees of the Santa Fe Railway System”:

In the olden days before good wages and fat back pay checks were unknown, yardmasters and their assistants were wont to wend their way over hills and vales to work with nothing to guide them but a lantern. But how times have changed. Night Yardmaster Clarence E. Lewis drives up to work in his Nash and then, when his day’s work is over, one will hear “Home, James, and don’t spare the gas.”

The earliest British-English use of home, James, and don’t spare the horses that I have found is from a satirical cartoon by Middleton, with explicit reference to a “popular song”—most probably Home, James, and Don’t Spare the Horses (1934), by the U.S. songwriter Fred Hillebrand (1893-1963).

Published for example in the Nottingham Journal (Nottingham, Nottinghamshire, England) of Thursday 3^rd January 1935, this cartoon depicts the British Conservative statesman Arthur Neville Chamberlain (1869-1940), then Chancellor of the Exchequer, as a lady telling “home, James! and don’t spare the horses!” to her driver, the tax collector; the horses are income-tax payers:

The Big Drive

WITH ACKNOWLEDGMENTS TO THE POPULAR SONG.

“HOME, JAMES! AND DON’T SPARE THE HORSES!”

[This week begins the annual Income Tax drive. £144 millions have to be collected by 31 March to complete the Budget estimate.]