The British-English phrase (there’s) trouble at t’mill seems to have originated in stage plays purporting to depict life in northern England, particularly in Lancashire, a county of north-western England, on the Irish Sea.

In this phrase, mill designates a factory fitted with machinery for a particular manufacturing process. Lancashire, and other northern counties such as Yorkshire, became major industrial regions during the Industrial Revolution.

Brian Saxton describes the context in which the phrase (there’s) trouble at t’mill occurs in Amateur stage: Are these ‘Northern’ comedies outmoded?, published in the Morecambe Guardian and Heysham Observer (Morecambe, Lancashire, England) of Friday 4^th December 1959—after describing what he calls “the seaside saga”, set during a bank holiday at Blackpool, a seaside resort in Lancashire, Brian Saxton writes:

Then there is the other type of ingenuous North-Country comedy, the industrial variety, which is set in a rather greasy dining room fitted with a large black kitchen range down-left which facilitates the drying of washing, the airing of clothes and the cooking of steak and kidney “pud” or tripe and onions (for some unknown reason, the authors of these ghastly plays believe this forms the Northerner’s staple diet).
In this sort of comedy, someone is always rushing in on such lines as “There’s trouble at t’mill,” or “Who’s coming t’club for a pint.”

The two earliest occurrences of the phrase (there’s) trouble at t’mill that I have found are from theatrical reviews by ‘R. P.’, published in the Manchester Guardian (Manchester, Lancashire, England):

1-: Of Tuesday 15^th February 1955—the phrase, in quotation marks, must have already been known to the readers, since the author did not feel the need to explain its meaning:

“GETTIN’ WED”
New Company at the Playhouse

What the public want is obviously hard to guess. The choice of the James Brennan Players for their first show at the Playhouse is “Gettin’ Wed,” by H. C. Leeson, which is, sure enough, a Lancashire dialect comedy with all the stock situations and is no guide at all, unfortunately, to what we can expect from them in future. The prospective bridegroom, looking naturally proper gormless, enters left in small bowler hat to give us the dramatic peak of the first act; and he is seen bathing the baby, in proper slapstick fashion, looking slightly grey with the passage of time, to give us all the sensation there is in the second. At the last we have violent indications of “trouble at t’mill” and the local gentry looking proper glum at the prospect of mixing with the people. That seems to be all there is to it. There is clearly not much scope for the company except to manage their accents and only Ray Mort, as the suitor, seems able to manage the Preston “r” and the Chorley groan ¹ with lifelike confidence. The audience reacted strongly enough to show that this is all right for those who like it.

¹ Preston and Chorley are two towns in Lancashire.

2-: Of Wednesday 18^th April 1956:

LANCASHIRE COMEDY AT LIVERPOOL
“As You Are”

Liverpool, Tuesday.
The Lancashire folk-comedy, one need hardly say, lives in a world of its own: completely static and, like Blackpool Tower ² and the Bury pudding ³, artistically far beyond reproach. Still: simple as it is, like those other North Country monuments, it takes a good company to engage an audience that is beyond enthralment by the simple “trouble at t’mill” theme. (This one is about trouble at t’underwear factory, a big advance.) William Stoker shows how to do it in his version of “As You Are,” by Hugh Mills and Wells Root, at Liverpool Playhouse. It is played at about twice the pace of the bumbling life it represents: it fairly knocks one’s flat cap aback. Battle-axe mother-in-law, weak but noble husband, sensible wife and the hire-purchase man whistle across the scene, leaving a bright and lively impression while hiding the creaking hinges of the action (though this is a particularly good play, as they go). Some cleverness in the acting also contrives to give it a strictly satirical edge: this could be developed.

² Inspired by the Eiffel Tower, which was erected in Paris in 1889, Blackpool Tower was opened to the public in 1894.
³ Historically part of Lancashire, now in Greater Manchester, the town of Bury is known for its black pudding.

The only early use of the phrase (there’s) trouble at t’mill that does not occur in a Lancashire publication is from the Western Mail (Cardiff, Glamorgan, Wales) of Tuesday 17^th February 1959:

Trouble at t’mill . . . so satisfying
By Harry Green

Club productions apart, “Any Other Business” is the most satisfying play put on at the Prince of Wales Theatre, Cardiff, for many months. George Ross and Campbell Singer, the authors, did a workmanlike job of dramatising a take-over bid—a topical subject when the play was written.
It is to the writers’ credit that the play survives a reduction in topicality, and stands as a good piece of intelligent entertainment at any time.
It is even more to their credit that they make accountancy exciting.
The heart bleeds for the directors who have a traitor on the board at the mill threatened by the take-over tycoon.
[…]
The tempo of the production is far too slow. Those heavy, deliberate Yorkshire accents put two minutes on every ten.