The British-English phrase pig in the middle, also piggy in the middle, designates:
– (literally) a ball game, usually for three players, in which the middle player tries to intercept the ball as it passes between the other two; this name also designates the player in the middle;
– (figuratively) a person, party, etc., caught between others in a conflict, dispute, etc.

Note: In the course of time, the name pig(gy) in the middle has been applied to several games, in particular to a game in which one player is encircled by others and must attempt to escape—cf., below, the quotation from The Folk-Lore Journal (1887).

The first three literal uses of pig in the middle, also piggy in the middle, that I have found are as follows:

1-: From Cornish Folk-Lore, by Miss M. A. Courtney, published in The Folk-Lore Journal (London: Published for the Folk-Lore Society by Elliot Stock, 1887) [Vol. 5, page 50]:

[For] “Pig in the middle and can’t get out” […], one of the children stood in the centre, whilst the others danced around him in a circle, saying, “Pig in the middle and can’t get out.” He replied, “I’ve lost my key but I will get out,” and threw the whole weight of his body suddenly on the clasped hands of a couple to try and unlock them. When he had succeeded he changed the words to, “I’ve broken your locks, and I have got out.”
One of the pair whose hands he had opened took his place and he joined the ring.

2-: From Hereford City Police, published in The Hereford Journal (Hereford, Herefordshire, England) of Saturday 10^th June 1893 [Vol. 180, No. 9,329, page 8, column 4]:

“Pig in the Middle.”—John Stainton, 68, Cotterell-street; Thomas Hiles, Clifford-street; Edwin Bagnall, 45, Whitecross-road; and Charles Richards, 65, Cotterell-street, were summoned for playing football in Cotterell-street, on the 23^rd May.—They pleaded not guilty. One of the boys remarked they were only playing “Pig in the middle”—(laughter).—The Head Constable thought the cases would be met by the defendants promising that the offence should not be repeated.—P.C. Matthews stated that the boys, with others, had a ball kicking about, and once it was kicked over into a garden. He was continually receiving complaints about the nuisance caused by the boys about the neighbourhood playing football.

3-: From a review of Games for the Playground (London: Longmans, Green, and Co., 1910) by J. S. Barker—review published in The Manchester Guardian (Manchester, Lancashire, England) of Wednesday 27^th July 1910 [No. 19,958, page 6, column 3]:

The twenty games explained, and illustrated by photographs that do not always catch their zest, are all capital, and any boy who grows up without knowing them is deprived of a part of his heritage. Neither the rules nor the nomenclature of the games are quite absolute, and “whirligig” is better known to some of us as stag, or even stagit, a variant hard to defend, which has certainly had some vogue in Lancashire. Whip is rather queerly changed into “trinaul,” and that noble game “pig in the middle” is here described as “kings.”

The earliest figurative uses of pig in the middle, also piggy in the middle, that I have found are as follows, in chronological order:

1-: From a correspondence from Bonn, West Germany, dated Sunday 20^th January 1952, published in The Manchester Guardian (Manchester, Lancashire, England) of Monday 21^st January 1952 [No. 32,838, page 8, column 2]—Konrad Adenauer (1876-1967) was the first Chancellor of the Federal Republic of Germany from 1949 to 1963:

Increasing concern is being expressed in Allied quarters here that differences of opinion between the Allied High Commissioners and the Federal German Government over the question of German financial support of Allied troops stationed in Germany will delay the signing of the new agreements which will redefine the status of the Federal Republic.
[…]
[…] It is recognised that Dr Adenauer’s obvious position in the time-honoured game of “Pig-in-the-middle” is becoming increasingly uncomfortable. He has now to engage in two tugs of war—with the Western Powers over occupation costs and with the Social Democratic Opposition over a possible revision of the Constitution.

2-: From a review of Hobson’s Choice (1954), a British film by David Lean (1908-1991), starring Charles Laughton (1899-1962), Brenda de Banzie (1909-1981) and John Mills (1908-2005)—review by Paul Holt, published in the Daily Herald (London, England) of Friday 26^th February 1954 [No. 11,840, page 8, column 2]:

WHAT is Hobson’s choice? Everybody says it, few know. Hobson kept a livery stable in Cambridge, and when undergraduates came to hire a horse, Hobson said: “You can have any horse in the stable, so long as it’s this one.” So began a great tradition.
The play sprang from the phrase and now David Lean, best film-maker in Britain, moves it on to films.
HOBSON’S CHOICE (Plaza) is a pretty comedy, pretending to be serious, hoping to be gay. An old drunk who runs a boot shop in Salford, Lancs, has three daughters and bullies them. How they revolt against his tyranny makes the fun.
How you will see it on the screen is a battle between Charles Laughton and a girl you know little of, Brenda de Banzie, the eldest daughter. They fight it out toe to toe and Miss de Banzie wins.
[…]
Playing pig in the middle is John Mills, the boot hand, who rises high at the hint of ambition given by the boot-master’s daughter.

3-: From Go on then, cheat the brute . . . If you can’t, then feed him instead, by Catherine Ross, published in the Evening Post (Reading, Berkshire, England) of Wednesday 10^th November 1965 [No. 51, page 3, column 3]:

Daughter loves salads. Husband insists only rabbits could be enthusiastic about lettuce. Husband likes meat. Daughter won’t touch it. Son likes coffee. Husband likes tea and beer. Daughter will have neither tea nor coffee. She hasn’t tried beer.
They can’t even get synchronised on an elementary thing like bread. Husband prefers wholemeal. Son likes soggy, steam-cooked, ready-sliced white stuff which mother would never even recognise as bread, let alone be proud of, and daughter sticks out for rolls.
I sit like piggy-in-the-middle ready to adjust to whoever is in for a meal. I’ve done so much adjusting that my ball-bearings are worn out.

4-: From a letter to the Editor, by Stewart Lyon, Chairman Disablement Income Group Charitable Trust, published in The Guardian (Manchester, Lancashire, England) of Monday 6^th October 1969 [page 10, columns 3 & 4]:

The personal tax allowances can be regarded as a form of social security for people at work, corresponding directly with the flat-rate national insurance benefits payable to people who cannot work.
Unfortunately under the present system the man or woman who works for a low wage is like the piggy in the middle, being eligible neither for a national insurance nor for the full tax allowances. […]
[…]
It is, of course, a natural reluctance to become the “pig in the middle” which encourages a small minority of people to live on supplementary benefits when they could be working. More seriously, it discourages the physically and mentally handicapped from taking up the kind of low-paid work to which many of them are limited because of their disability.