The whole idea of a world war in 1862 is unsustainable, especially over Trent. If the Union do not yield to British demands as they did in our-time-line, they alone will be at war with the British Empire and absolutely no one in the whole world has any reason to come to their aid. They will be fighting a war it is almost impossible for them to win.
The worst case scenario for the Union is that France not only extends its adventures in Mexico but joins in the war against the them as well. Such an alliance means that the British can fight any combination of enemies on the high seas with a very high confidence of winning. This is because in this period the size and power of the British fleet was based on the need to defeat a combination of the second and third most powerful navies in the world. In this scenario France is the second most powerful navy after the Royal Navy and Russia is the third but 1862 it can easily be blockaded in the Black, Baltic and White seas. The navies of the other nations are so far behind the Royal Navies that even in a total combination they cannot offer any significant resistance to the Royal Navy especially if the French help the British. In 1862 with Britain at war with the Union nobody with a sea coast is going to go to war with Britain.
In 1863 the situation is different and, as I have said in my previous post, there is more chance of other nations becoming involved in a war in North America. If France and Britain were to support the Poles of the Russian Empire in the January uprising or even more unlikely go to war in their cause then Russia might activate its commerce raiders in New York and San Francisco. If this happened then the British might go to war with the Federal Americans if they were to harbour the commerce raiders in a way the British considered to be contrary to international law. Similarly the French might take the opportunity offered to send more troops to Mexico but they would be very cautious in doing this if they felt they might need those troops in the Dardanelles to protect the Ottoman Empire and the Bosporus.
The British will not ally with the Confederacy although they will work with them, the French might. Interestingly the Brazilians have a half decent navy and are very angry about the behaviour of Union cruziers, they might ally with the Confederates and the British, if they are receptive.
Having said all of the above, Russia would probably be looking for a way out if France and Britain declared war on her. The memory of the Crimean war is still very much in the minds of the government and military. The emancipation of the serfs is causing huge societal and economic problems and they are also in the closing stages of long war against the Circassians and Caucasians. In short they are not in a good position to fight a war abroad.
The British are also unlikely to want to push the war against the Union any longer than they have to. In 1863 they are over extended. They are fighting the Maori in New Zealand, occupying Lagos Island, bombarding house Satsuma at Kagoshima, peripherally involved in the Taiping, skirmishing on the Bhutanese border, there is drought in Jamaica and they are fighting the Ashanti.
As for the rest of the world in 1863 many countries are mired down in a series of local disputes which it would not be easy to globalise.
In Imperial China the Taiping rebellion is entering into its final bloody phase.
The Risorgismento has not yet run its course in the Italian states, although the fighting is over for a while.
Ecuador and Chile are at war.
Japan in entering turbulent times towards the end of the Bakumatsu with both the bombardment of Kagoshima and the naval battle at Shimonoseki
The Uruguayan Civil War is in full swing.
The Spanish naval expedition to South America is involved in the Talambo Incident which led to the Chincha Islands war the following year.
1864 has the greatest potential for world war as there are so many regional conflicts going on; in addition to the above the war of the Triple Alliance breaks out, so does the Danish-Prussian, the Cincha Island war and the Duar war starts. However, there is no one unifying event, certainly not the American Civil War that could bring all of this conflicts together into a single war.