So I can only really answer your question in regards to Britain and France. These two powers were the closest to adopting intervention policies and therefore are going to be the most interesting to look at.
The official stance of both countries, and of all European countries, was that of neutrality. But that does not tell the entire story.
The Confederacy realized the importance of European powers immediately and sent a delegation to Europe with the hope of gaining support. France and Britain both heavily relied on Southern cotton for their textiles, and the Confederacy planned to use that reliance to their advantage. In France the delegation was not officially received, but they did meet with Napoleon III to try and convince him to officially recognize Southern independence and help break the Union blockade in exchange for cotton. Napoleon III was swayed and at first pushed his government to recognize the Confederacy, but the French Foreign Minister helped talk him down and after the US took New Orleans in 1862 there wasn’t much support for intervention.
Another interesting aspect for France though is that Napolean III was seriously exploring the idea of colonizing Mexico. The US was very much against this action though, while the Confederacy claimed they would not mind. The US winning the war put a serious damper on the idea and was ultimately abandoned altogether.
Frances role was relatively minor. Although some of the wealthier classes in France backed the Confederacy through some loans, the French government never really supported the Confederacy in any way and actually blocked the sale of some ships intended for the South.
Britain was a bit more complicated. Their interest in Southern cotton was stronger and they came much closer to actually going to war with the US or officially recognizing the Confederacy.
Once again the upper class in Britain, like in France, was in favor of Southern independence. Their fortunes often relied on cotton and they wanted to maintain that relationship and hopefully shorten the war before supply ran dry. Those in the British government were mixed on the issue. Most, including the Prime Minister, mostly favored neutrality.
One of the issues for Britain was how to recognize the blockade on Confederate ports. These ports were what supplied Britain with cotton, but if they were blockaded by the United States, then Britain would have to officially obey that blockade under international law. However, in doing this the British government also had to officially recognize the Confederacy as an independent country, since blockades of your own ports was not allowed under international law. Those in the British government who wanted Southern independence brought up these points. Also pointing out for a blockade to be real and recognized, it needed to be effective. The Confederate delegates to Britain claimed the blockade was not effective because ships still got through. In the end though after talks with the between American ambassadors, Lincoln, and British officials, they convinced Britain to not officially recognize the Confederacy. The British government only recognized the Confederacy as a “belligerent state”, which did not help their cause.
The British government ultimately had bigger issues than supporting the Confederacy. Costs of the Crimean War we’re still fresh on their mind and other powers in Europe had potential to threaten Britain. By the time the Union won the battle of Antietam and Lincoln gave his Emancipation Proclamation, support in the British government for the Confederacy was almost nonexistent. Supporting the Confederacy after the Emancipation Proclamation would make it look like the British were specific fighting FOR slavery. The British government would stay officially neutral for the rest of the war.
Private investors in the UK did support the Confederacy though. Iron clads and ships were built in British ports from private investors in agreement with the Confederacy before being sailed to the South. The US government was furious at this blatant disregard for the official neutrality though and wanted the British government to do something about the ships being built. Eventually they would crack down on the ships leaving port, but British officials often looked the other way. The same goes for weapons. Thousands of Enfield Rifles made their way secretly to the Confederacy even very late in the war because of private financing in Britain.
One other important event in the Civil War involving Britain was the Trent Affair. The Confederate delegation to Europe sailed to the Bahamas where they boarded the British mail ship RMS Trent. From their the ship planned to bring them to Europe. However, the American war ship USS San Jacinto knew the delegation was on the Trent, and after the Trent left port the San Jacinto intercepted them and forced them into being boarded. The Trent was then forced to sail to Hampton Roads, VA where the delegation was arrested. The British government was furious about this. They felt the US had no right under international maritime law to board a ship flying a neutral flag. The US claimed the knowledge that an enemy delegation was aboard was enough reason to board legally. The Trent Affair brought Britain and the US very close to war with each other. It was only after Lincoln apologized and the Confederate delegation was released and allowed to keep sailing to Europe. After this, public opinion in Britain swayed back to staying out of the war.
Sorry if it feels like I rushed through these. Although it’s not really well taught in history classes, there were actually a lot of things that happened between the US, Britain, France, and the Confederacy. I tried to touch upon as many as I could. If you have any questions or want more information on specific things I mentioned or people that were involved feel free to ask.
For further reading I highly suggest James McPherson’s book War on the Waters. Since most of the international stuff involved ships, this book is a great resource and where I got most of my sources.