Was there ever any significant chance that Britain would intervene in the American Civil War to assist the South?

There was, though the cotton famine on which Confederates pinned their hopes was ultimately a secondary consideration in British political maneuvering. This was due to a number of countervailing factors that undercut the political impact of the cotton famine.

That Europe was severely impacted by the sudden disappearance of American cotton is true. By the onset of the Civil War, the United States (read: the South) produced 77% of Britain's 800 million pounds consumed and 90% of France's 192 million pounds consumed annually. Ironically, the first interruption to Britain's cotton supply was not the Union naval blockade - which until 1862 to become effective - but the South's self-imposed embargo as they tried to trigger an economic crisis in Britain (which didn't work, due to British cotton reserves). By the time they realized how badly they needed cotton revenues, in 1862, the blockade had ramped up and tonnage leaving the Confederacy had dropped precipitously.

By 1862, this did have significant impacts on British industry and social stability (from McPherson's Battle Cry of Freedom):

In July 1862 the supply of raw cotton in Britain stood at one-third the normal level. Three-quarters of the cotton mill workers were unemployed or on short-time... [Son of American minister to Britain] Henry Adams conceded as early as May 1862 that "the suffering among the people in Lancashire and in France is already very great and is increasing enormously." Chancellor of the Exchequer William E. Gladstone... favored British intervention to stop the war and start the flow of cotton across the Atlantic.

So what were those countervailing factors?

In the short term, there were two major disincentives to British intervention. First being that they wanted to avoid antagonizing the northern United States, which was a major source of foodstuffs and a major consumer of British manufactured goods. This concern was especially pressing in the light of episodes like the Trent Affair, where war with the Union seemed like a very real possibility. The second was Britain's moral stance against slavery - most of the political elite and population as a whole found the practice abhorrent, and any political moves in support of slavery was a non-starter. This wasn't an issue when the Union merely fought against secession, but once Lincoln issued the Emancipation Proclamation, the Union took the moral high ground and made support of the Confederacy politically unpalatable.

In the long term, what worked against the King Cotton strategy was the simple fact that United States was not the only supply of cotton for Europe. European powers were able to reallocate resources to places like India, Egypt, and Brazil to recover their cotton supply. By the end of the war, British cotton consumption had returned to prewar levels due to these sources (as well as resumed cotton trade from Union-controlled territory).

Ultimately, what mattered for British intervention was not cotton, but their success in the war. Prime Minister Palmerston held that the Confederacy had to establish their "separate independence is a truth and a fact" through military success to earn British recognition and mediation. Accordingly, the peak chance of intervention was during Lee's 1862 Maryland Campaign. The Union was still recovering from the catastrophe of the Peninsula Campaign, and abolition of slavery had yet to become a war goal for the North. Palmerston was ready to concede to pro-Confederate forces in the British government, on the condition of a decisive Confederate victory.

Whether that was ever a possibility is hard to know. That would come down to the well-known what-ifs surrounding the Maryland Campaign and the Battle of Antietam, particularly the Lost Order. But as it happened, McClellan won his shaky victory there, directly leading to Lincoln's Emancipation Proclamation (he promised to himself that he would take victory as a sign from God that he should issue it). The demonstration of the Union's sound military footing, combined with abolition as a war goal, ended any real chance of British recognition for the remainder of the war.

Sources:

James McPherson, Battle Cry of Freedom
Sven Beckert, Empire of Cotton