Of American-English origin, the phrase when (also if) push comes to shove means when a situation reaches a critical point and one must commit oneself to an action or decision.

The image is of having to shove when mere pushing is ineffective.

The earliest occurrence of the phrase when (also if) push comes to shove that I have found is from An Interesting Story of What a Girl Found to do and How She Succeeded, in the column The Home Circle, by ‘Bessie’, published in The Weekly Floridian (Tallahassee, Florida) of Tuesday 5^th February 1889:

On arriving at the town which she intended making her new home she found everyone excited, and the men were all seemingly on the ragged edge of despair. Upon inquiry as to the cause of the excitement, she was told that the only barber of the town had packed up his outfit and skipped by the light of the moon the night before, owing nearly everyone in the town, but that the main cause of the excitement was that the male citizens were in despair because they would have to do their own shaving until they could find another barber. Her heart throbbed wildly; her brain became a busy workshop; why could she not take advantage of this splendid business opening; she had for years shaved her father and cut and trimmed his hair, and he had often told her that she was an artist in that line. She felt inspired and determined to make the venture. Without stopping to find a boarding house she hunted up the agent who had the vacated barber shop in charge, and nearly paralyzed him by taking a year’s lease and paying the first quarter’s rent in advance […].
[…] She fitted up in elegant style, for that town, and the first day she opened had such a steady run of custom that when night came her arms ached. She fitted up a private tonsorial parlor for ladies, and all the girls of the town came to have their hair banged. The city editor of the town paper gave her a two-column write up, and called her an “apterous tonsorial angel.” She secured two bright young girls from the city as apprentices, and it was not long before she was running three chairs and had all the business she could attend to. Young men who had never before thought of shaving dropped in regularly to have their faces scraped, and if they were going to a ball or party always came to her tonsorial parlors to have their hair dressed. It is needless to add that she accumulated much wealth, for with such a run of trade that fact is very evident. If this were a novel it would not be complete without marrying her to some prince of the blood, but this is simply a business illustration, to show what a girl can do when push comes to shove.

Either a misprint or an early variant occurs in the following from Are We Returning to Primitive Methods?, published in The Macon Telegraph (Macon, Georgia) of Saturday 13^th February 1897:

We are told by a newspaper that echoes these tales of distress, and which has a penchant for war, that these senators “are more than a match for British diplomats.” But, “if pinch comes to shove,” as old Sol was wont to say, will these gentlemen put on the habilaments of war and prove “more than a match” for British ironclads or Spanish machetes?

The phrase reoccurs in The Macon Telegraph (Macon, Georgia) of Monday 28^th February 1898, in a series of short paragraphs about the events that were leading to the Spanish-American War (April to August 1898)—the ‘yellow newspapers’ * were campaigning for a war with Spain by claiming that the Spanish were responsible for the explosion, on Tuesday 15^th February 1898, of USS Maine in the port of Havana, Cuba:

When “push comes to shove” will editors of the Yellow Kid organs enlist?
Governor Johnston’s voice is for war. Will the governor enlist?
War is a good thing push it along.—Yellow Kid.

* The specific application of yellow to journalism originated in the name of a cartoon character, The Yellow Kid (a child wearing a yellow shift), which first appeared in the New York World newspaper in 1895. When the originator of the character, R. F. Outcault, was persuaded to transfer his popular cartoon strip to the New York Journal in October 1896, the proprietor of the New York World, Joseph Pulitzer, commissioned another artist to produce a cartoon strip featuring a character with the same name. Subsequently, yellow kid had a brief period of currency as a means of referring to the two rival newspapers, and to the methods used by them to attract readers.

The phrase then occurs in The Morning Post (Raleigh, North Carolina) of Sunday 16^th June 1901:

STOCK COMPANY CHARTERED
Windsor Cotton and Peanut Company Has a Variety of Powers

The Windsor Cotton and Peanut Company, of Windsor, N. C, was chartered by the Secretary of State yesterday. The capital stock is $25,000, and the incorporators are R. C. Bazemore, Geo. L. Mardre, Jas. J. Mardre, Francis D. Winston, Thos. Gilliam. The company is authorized to deal in real estate, buy and gin cotton, buy and clean peanuts, make peanut meal, construct and operate an electric light plant, operate saw mills, manufacture furniture, deal in horses, mules, and other live stock, manufacture or buy and sell harness, and vehicles, buy and sell corn and wheat, and grind and sell product of same. In fact the scope of the company’s privileges are such that in the language of a gentleman who read the articles of agreement, “they can constitute and manage the enterprising town of Windsor ‘if push comes to shove.’”