One fact I find interesting is that, had Britain and America gone to war in the 1850s under Palmerston’s leadership, he would have had none of Lincoln’s hesitancy in making it a war of liberation. “If we are weak in Canada, the Americans are still more vulnerable in the slave states… A British force landed in the Southern part of the Union, proclaiming freedom to the blacks would shake many of the stars from their banner” (Palmerston to Panmure, 24th September 1855)
In Lagos in 1851, they depose the king and install a new one opposed to the slave trade; in 1852, King Gezo of Dahomey is forced to abolish the slave trade and export palm oil instead. It’s not limited to Africa either, as recognition of the Republic of Texas and an independent Brazil is made conditional on abolition of the slave trade. In June 1850, Royal Navy ships start sailing into Brazilian harbours and seizing suspected slavers, fighting pitched battles with Brazilian forts and troops in order to do so.
Whenever Britain has attempted to do something about slavery in the United States, it’s been confronted with the whole strength of the Union (the best example being the 1858 boarding dispute). If the Confederates win, the dynamics of power change. Britain then has a strong anti-slavery North with no love lost for the South, and a weak slave-owning South which is far easier to bully. The idea that British policy would remain the same, or that British "moral influences" would have the same limited effect, is fairly implausible.
Given what has already happened and what we know the British were planning, within a year or so of Confederate independence Britain would have the right of search over both Union and Confederate vessels and there would most likely be an anti-slavery patrol off the coast of Cuba. More speculatively, if the Union attached a ship or two to this squadron, a dispute could quite easily arise from them boarding a Confederate flagged ship- something an unscrupulous Northern President looking for an excuse for a second war might be quite happy to take advantage of.