There's a lot to unpack here but I'll take a stab at it in a brief summary; there's enough going on to fill several books (which have been written). To put it briefly, Britain and France considered attempting to influence the outcome but ultimately decided not to try. I would argue that the modern countries that most influenced the outcome are Ireland, Germany, Egypt, and India.

When the war began, the U.S. government made clear that all foreign intervention would be treated as an act of war. A blockade of southern ports was also declared but a blockade must be both declared and enforced to be legally binding so for a while there was some wiggle room as the Navy was ramped up.

Britain was on the fence about the whole thing, for a variety of pretty complex reasons. They didn't like the slave trade or slavery, which they'd largely banned in 1807 and 1833 respectively. Elements just flat out didn't like the U.S. as well and saw the war as useful as long as it kept killing Americans (and the Irish, more on that later). Nevertheless, the U.S. was a huge exporter of food to Great Britain so there would be serious consequences to intervention and the U.S. was also a huge British market for manufactured goods. The British used blockades as a foreign policy tool themselves quite a bit so they weren't really inclined to challenge the validity of the Union declaration early in the war and by the later stages the U.S. Navy was objectively enforcing the blockade.

There were two points where Britain almost intervened. The first was the Trent Affair, where a Navy warship removed two rebel diplomats from a British mail packet in 1861. There was a major diplomatic row that was addressed pretty quickly but it drove home to the British that any war with the U.S. would probably lead to the permanent loss of Canada due to the massively expanded, battle hardened U.S. Army available from about 1862 or 1863 onward and the increasingly formidable U.S. Navy which was capable of fielding a substantial fleet of ironclad warships in their own coastal waters, not to mention an increasingly robust blue water Navy.

The second was the issuance of the Emancipation Proclamation, which France feared would lead to a race war in the South and thus approached Britain about leading a humanitarian intervention to stop the bloodshed. Britain was not interested at that point. The Union army had grown almost immediately from 16,000 or so when the war began (a manageable number for the British forces in Canada) to over 500,000 within a matter of months in 1861. By the issuance of the Emancipation Proclamation in 1863 that army was battle hardened (having fought some of the bloodiest battles of the war like Shiloh and Antietam already and with key commanders like Grant, Sherman, and Sheridan emerging) and would be nearly impossible to prevent from seizing Canada, not to mention almost impossible to dislodge once they had.

The rebels made yet another fatal blunder - they assumed that cotton would tip the scale. What they didn't realize was that Britain had a large supply of cotton already, about a year's worth, and was developing alternative sources in Egypt and India. The South's trump card turned out to be worthless. Doubling down on their stupidity, southern cotton interests then began an unsanctioned boycott of British buyers thinking they would force Britain's hand. Instead they undermined their own diplomats and drove British buyers to more urgently find other suppliers.

France had a variety of competing interests that favored intervention but wasn't willing to make a move without Britain. In fact, two Orleans princes fought for the Union army. France had a colonial experiment going on in Mexico that was much more favorably viewed by the South than the U.S. government and had other ambitions in the area. France was also hurt much more by the lack of southern cotton. Private interests invested in blockade runners but that was about as far as anything went other than the suggestion about intervening after the Emancipation Proclamation. Lack of British interest and a looming war with Russia over a succession crisis in Greece put that idea to bed. Dan Doyle argues in Cause of All Nations that Giuseppe Garibaldi's endorsement of the Emancipation Proclamation and the Union cause also helped tilt the scales in favor of non-intervention.

Fearing war with Britain, France, and the Ottomans, the Russian navy sent two fleets to American waters. At the time this was seen as support for the Union but it appears to have been to avoid the fleets being trapped if war broke out. They also attempted to construct a telegraph line from Alaska, through British Columbia, and to the U.S.

The countries that had the greatest influence by far were Germany (or the patchwork of German states at the time) and Ireland, who each supplied about 200,000 of the Union's 2.2 million soldiers (although a substantial number of Irish fought for the South as well), which about the same number as the 200,000 or so emancipated slaves that fought. About 25% of the Union army was foreign born but those were far and away the largest single contingents (there was a Polish legion, tens of thousands of French Canadians, British, Italian, and Canadians too, nearly 100,000 Dutch; the list goes on and on). It's possible the war couldn't have been won at all without the support of these men who helped bear a burden of killing and dying that might have been too high for the native population to bear alone. The war to preserve the Union was about as much of a global effort as one can conceive of in the modern day and age.

That's just scratching the surface really.