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The Brits pretty much gave up on interfering with USA after the War of 1812, which demonstrated pretty thoroughly that it wasn't going to work. As the US got more powerful, they also had a pretty obvious hostage in their Canadian colonies. Not a good idea to poke the tiger.

By 1860 the vast majority of Brits opposed slavery, and so favored the Union.

Probably true, though I'd say majority, not vast majority.

The Union, however, was quite unpopular among the upper classes, who still ran things in UK. They would have been quite happy to find an excuse to recognize the CSA or even to interfere militarily. The slavery bit was the only thing that kept them from doing so.

Which means that if independence had really been more important than slavery to the South, they could have had it. Abandon slavery, and their freedom would have followed almost automatically.

The problem, of course, is that preserving slavery was the reason they wanted independence, which would have been meaningless for them without slavery.