It depends on how it is achieved - a hands off internal collapse of the Union, whether TL-191 style, through an 1864 Lincoln loss, or some other way, is a completely different scenario from one where the UK gets drawn into a North American conflict and the world is forced to digest what such a major shift in the balance of power means. A purely internally driven Confederate victory has the most potential for a minimal impact outside of the issue of slavery and its image in public opinion, in Europe and elsewhere. Other ties between the CSA and Europe would have to develop on their own out of European interests and initiative. On the other hand, if Britain intervenes on the side of the CSA for whatever reasons and this policy is vindicated, someone has already mentioned how this could embolden British confidence to simply employ a big stick in pursuit of its interests around the world, including in North America. Put no stock in the idea that the UK and US will inevitably be chums. In this world, both sides have every political incentive for hostility, regardless of the fact that certain close economic ties will continue in the world of money people. Daily trade is not the end-all be-all of a nation's economic interests, and that didn't stop WWI, WWII, or the current US-China deterioration. Britain won't be going to war to take California, but during a Trent war over Canada, British leadership may get fanciful ideas of splitting the US apart, such as the Northwestern Confederacy that the Confederates and some northern copperheads put their faith in, but which had negligible support. They'll find they are not able to project that kind of power into the heart of the US.
One fallout would be the USA's first "special relationship" - it would not be with the UK or some other democratic nation, but with Russia and/or Germany. It's killing far too many butterflies for my comfort to speak even of the First World War with an 1860s POD, but one can see the political cartoons with Britannia fending off the three eagles.
A France that remains occupied in Mexico will likely not entertain needless wars with Prussia. Germany will have to be united some other way. It would be interesting to explore a gradual German unification that takes decades and endures setbacks. Same for Italy, if it can be kept out of Rome (and Veneto).
It would be interesting to see what kind of influences and frictions there would be with a US-Russia relationship. It may or may not work, and it may do things like bring the Polish question to awareness in the US or bolster the influence of Russian liberals and constitutionalists.
It's not clear that American democracy as we understand it would survive a civil war defeat. It's always just kind of assumed that a US with a heavily militarized politics and the "bigger government" that brings, nationalist revanchism, likely with a greater police state presence in everyday life, and engaging in the global power game with hostile nations on each border just continues on constitutionally the same. There are no American Boulangers in any timeline I'm aware of. It's amazing that that isn't explored more. Maybe the current political climate in the US will change that. That has huge implications for European, and world history.