Henry John Temple, 3rd Viscount Palmerston (Prime Minister throughout the entire war) sums it up pretty well himself;

... it is no doubt certain that if the Southern union is established as an independent state it would afford a valuable and extensive market for British manufactures but the operations of the war have as yet been too indecisive to warrant an acknowledgement of the southern union.

The British Conservative establishment was more than willing to throw their weight behind the Confederacy for the most part. Southern cotton without high tariffs? Check. The weakening of Britain's only major rival in North America? Check. An independent South was a British wet dream. The British government would happily have backed the Confederates if they looked likely to win.

By October 1862 in the wake of repeated Confederate victories Palmerston's administration was on the brink of attempting to mediate a peace between North and South, with the Foreign Minister (John Russell) writing to Palmerston on the 4th;

I think unless some miracle takes place this will be the very time for offering mediation.

What changed everything was the Battle of Antietam and the collapse of Lee's Maryland campaign. Not only did these events rock British faith in a previously invincible Confederate military, it gave Lincoln the confidence to announce the Emancipation Proclamation. That announcement changed the war from a regional dispute to a war between freedom and slavery in the eyes of international observers. And as /u/Keiji99 has covered, the British public was pretty much opposed en masse to slavery.

And by the 24th with all this having had time to set in, Russell writes to Palmerston again stating;

As no good would come of a Cabinet [to consider mediation], I put it off.

 

Sources:

'Great Britain and the American Civil War', Adams, E.D.

'Lord Palmerston', Ridley, J.

'Basic History of the Confederacy', Vandiver, F.E.