Answering the question more generally, without addressing any of the long running debates and a disagreements, there are a number of hard points any foreign intervention in the ACW is going to hinge around.
1. Almost no-one seriously wants the Trent incident to turn into a war. Lincoln has one war. The British are coming off of a major and serious revolt in the jewel in their crown. Both powers prefer unobstructed sea lanes and the status quo of peace. The US would prefer not to manufacture everything at home; the British like grain coming in. The British would rather keep getting rich and expanding the areas they control at peace. Wartime armies are expensive, especially when...
2. The North Atlantic is a Big, Wide Ocean. Goes without saying, but its something to consider. Compared to the Wars of the French Revolution, armies need more. More bullets, more powder, more shot and shell. Warships can't elegantly sail as long as they have food supplies, coal is bulky, heavy, and gives warships short legs. Sealift takes a lot of ships. Very little can match the railroad in that day and age. This affects the use of...
3. The British Army is Quite Good, but has a list. In 1861, the U.S. has a lot of good engineering and staff officers, but for how to use a regiment of rifles, the British have an edge, and doesn't need to unlearn as much. Regiment for Regiment, the British have an edge. They have fewer regiments though, and likely less ability to expand quickly, and again - they also have to hold down a good sized chunk of the globe. AS referenced in (1), if the choice is between India or Boston, the choice is India. Bleeds into...
4. Time is very much on the Union's Side. Every passing month is more for the USA. More indigenous production of war material, more tactical competence, more competent men in high command, fewer idiot illusions. There's a faction that views the ACW U.S. Army as just another tribal society to be swept aside by a few volleys by the redcoats; but one remains skeptical. Each month is also another month where British merchant shipping is not hauling resources to the UK and finished goods to market. Again, all for an avoidable war of choice, dealing with...
5. The American Civil War Was About Slavery. The myth making of the war not being about slavery doesn't exist until after the war. The South needs to lose, and abolition needs to be achieved, for the South to need another tale of what it was really about. Now honestly, I don't think this is as big a bar to British intervention as others do. Chattel slavery is horrible. But read up on British India to see how little of a shit the British cared when the rubber hit the road.
Still though, you can argue that the British cared a great deal about putting the veneer of progress over the reality of colonial exploitation. Supporting a band of feudal throwbacks does not do this.
You can make a tale where the Union still succeeds; you can make one where the some segment of the South obtains some measure of independence - likely as a British client, as has been mentioned above. Heck, you can make one where the British regain their colonies and the British run their Empire worldwide for all time - but it's a heavier lift.
Still though, those five need answer.