I doubt if Americans required British intervention to foment disunion here -- we had plenty of disunionists, aka "Fire Eaters", without British help.
The historical narrative leading up to US Civil War has been related many times by many different historians, none of whom emphasized a key role for British subversion.
By 1860 the vast majority of Brits opposed slavery, and so favored the Union.
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.