There were multiple times were the US civil war threatened to become a world war between US & Russia vs England & France & the CSA. I've read a lot on England dispatching troops to Canada, France and Spain's interventions in the new world and Russian and British fleet movements. However, I have not seen anything about the armies in Europe preparing for such a fight on the continent, even after Russia made it clear they would go to war if Britain and France recognized the CSA.
Were there any troop deployments or military construction on land in Europe to prepare for this? What about neutral countries that had a high chance of being brought into the conflict by geography, such as Prussia?
It might be far better to view the US Civil War as a sideshow for the European powers. Except for the British troops sent to Canada, the Russia/UK/France/Prussia posturing was all about Europe and their relative positions in the overall strategic position. ( As for the UK, a grand total of 11,000 new troops were sent to Canada and that was all the UK had available. The rest were tied down - mainly in India and other colonial messes).
The real reason the Russians deployed their fleet to US waters and were willing to declare war on the UK and France if they intervened in the Civil War was that Russia felt that it was the best chance at hitting back and recovering some of the losses from the Crimean War. Tie the UK and France down with a 2 ocean war on 4 continents where Russia and the US could be able to apply local superiority of force at really any point. Russia didn't particularly care about the United States' side, per se. They did have good relations with the United States, but aside from a few specific items, had no real stakes in the outcome of the Civil War.
The real flashpoint was Poland and tensions were incredibly high in these years as the shifting power balance in Europe meant that several powers were looking at a different distribution of resources and the Poles had launched their own revolution looking for freedom. Russia was viewed as being weakened by the Crimean War and was starting social reforms, such as freeing the Russian serfs in 1861. The timing of the Russian fleet to the US was Russia covering its options by moving its fleets to warm water ports where the fleet would not be frozen in and be unavailable at the start of a war.
There was the possibility of a world war in 1863 during the January Uprising in Poland as various sides lined up to support the Polish rebels or back Russia and Prussia. And, the Polish rebels were trying very hard to make that a military intervention happen. There was definitely a move to pull forces back to Europe as available for all those powers. But, none of the powers seem to have seriously considered intervening on the Polish side due to going to war against both Prussia and Russia. It was certainly possible had things played out differently. Overall, this was closer to a reality than the UK/France deciding to militarily intervene in the US Civil War.
Spain was staying as far from involvement in any of these messes as humanly possible. Except for a joint intervention at the tail-end of the Mexican Civil War in 1861, they didn't want to get involved in either the Civil War or Polish question.