It may be marginally true that linen slightly replaced cotton in Britain during the US civil war, but it was not a significant factor.

Britain was starved of cotton during the war, and 440,000 Lancashire cotton mill workers were thrown out of work, lost their wages, and were desperate and starving. It is called the 'Cotton Famine' in British history.

The Confederacy had high hopes that Britain might come in to the war on their side. They thought that cutting off the cotton, on which the textile mills of northern England depended, would bring Britain in to the war on the Confederate side to restore supplies.

The Confederates underestimated the moral opposition to slavery in Britain, which made support of the Confederate cause by the British unlikely.

One of the most principled and courageous stands against slavery and in favor of the Union came from the mill workers of Lancashire in England. It deserves to be better known in the USA than it is. In 1860 there were 2,650 cotton mills in Lancashire, employing 440,000 people. As the Union blockade, (and a Confederate embargo, aimed at bringing Britain into the war on their side) cut off the cotton, the mills shut down and the workers lost their jobs.

Despite hardship and starvation, in 1862 a meeting of the increasingly desperate mill workers still resolved that they would support the Union and suffer if it led to the ending of slavery. (Source: Sir Arthur Arnold, “Cotton Famine 1961-1864”, 1864) They sent this letter to President Lincoln:

"... the vast progress which you have made in the short space of twenty months fills us with hope that every stain on your freedom will shortly be removed, and that the erasure of that foul blot on civilisation and Christianity – chattel slavery – during your presidency, will cause the name of Abraham Lincoln to be honoured and revered by posterity. We are certain that such a glorious consummation will cement Great Britain and the United States in close and enduring regards."

—Public Meeting, Free Trade Hall, Manchester, 31 December 1862. (Source: http://www.manchester.com/features/manchesters.php)

Lincoln sent the following letter back:

"Under the circumstances I cannot but regard your decisive utterances on the question as an instance of sublime Christian heroism which has not been surpassed in any age or in any country. It is indeed an energetic and re-inspiring assurance of the inherent truth and of the ultimate and universal triumph of justice, humanity and freedom.

I hail this interchange of sentiments, therefore, as an augury that, whatever else may happen, whatever misfortune may befall your country or my own, the peace and friendship which now exists between the two nations will be, as it shall be my desire to make them, perpetual."

—Abraham Lincoln, 19 January 1863 (Source:http://www.bbc.co.uk/news/world-21057494)

Lincoln sent more than just kind words. On 9 February 1863 the ship George Griswold, docked in Liverpool. It was sent by Lincoln, aided by the citizens of Philadelphia and New York, and carried bacon, bread, rice, corn, and 15,000 barrels of flour to relieve the suffering of the Lancashire mill workers.

(Source:http://revealinghistories.org.uk/tpl/uploads/Education-card-4.2.pdf)

Although some elements of British society might have been tempted to support the Confederacy, and although Union bungling (in the Trent incident) came close to enraging the British and bringing them in on the Confederate side, Prince Albert personally got involved to calm things down and keep Britain neutral. 440,000 principled and courageous mill workers were part of the reason why.