I quote Wikipedia: "
In September 1862, during the Confederate invasion of Maryland, Britain (along with France) contemplated stepping in and negotiating a peace settlement, which could only mean war with the United States. But in the same month, US president Abraham Lincoln announced the Emancipation Proclamation. Since support of the Confederacy now meant support for slavery, there was no longer any possibility of European intervention.[88]". So if Britain, France and Spain want to step in, they need to do so before September 22 1862.
I quote my first chapter:
Between May and September of 1862, the forces of the Pact of London took Southern Mexico and most of Central Mexico. It was at this time that the Second London Convention took place.
So on this Convention, the three powers can decide whether or not they intervene on the ACW, and, because of their troops on the American continent and their fleets on the Pacific, they wouldn't need much time to mobilize.
As ITTL Maximilian would become Emperor by late 1862, bringing with him support from Austria, and at the point of his coronation he has under his reign Southern & Central Mexico, plus the French haven't had as much casualties as IOTL, as the Invasion of Mexico was rather smoothly executed. So, by late 1862 Austrian troops begin arriving to Mexico, the French army is more robust, Maximilian I has more support over Mexico and the London Pact has a lot of fresh troops plus a formidable navy on the Western Hemisphere. So what they just need is to attack on the correct time and hope the elites at home aren't too angered by the London Pact's support for the CSA. But to rally as much of the elites as possible, the London Pact needs to attack before the Proclamation of Emancipation (the one issued 22 September 1862).