it is an interesting question and the effect of such a conflict in the short medium and long term.
I think one of the short term reason the British did not was food.
After the corn laws were repealed the British were very dependant on imports of wheat from the Union and the harvest in europe at the time were bad leading to hunger.
I think the British needed Union wheat more than they need Confederate cotton.
Union diplomats seem to have done a better job than the Confederate ones.
Assuming the British and French join the war on the CSA side what do they gain?
Access to Cotton, tobacco,Sugar,Naval stores and Rice from the CSA
Tariff free access to CSA markets for their industrial good and weapons
They also limit the size and power of a potential rival to its empire.
How would in help the CSA?
The biggest effect would be the economy with the ports open this might help control inflation.
The British have the printing presses to forge Union green backs on a large scale to disrupt the Union economy.
How would the Union hit back?
I think the US could build commerce raider to disrupt British and French ships world wide and drive up insurance rates on Lloyds of London.
There there was even an implied threat that the Union could build commerce raider for the IRB(The Fenians) and crew the with Union sailors.
Assuming the CSA can win with this help how do things change?
Bad relations with the USA and possible a arms race with Navy on both sides.
America might aid rebel groups in the British empire like in Ireland and South Africa etc. British might have a lot more problem in South Africa and Ireland.
Catalpa rescue this could lead to further tensions
https://en.wikipedia.org/wiki/Catalpa_rescue
In WW1 the British might think twice about joining the war as they may not get help in loans or USA joining war on their side.
If the British do join the war they might need to come to an early settlement with the CP maybe 1916.
Short war may mean no communist in Russia and no big loans for the USA so the default on war loans form Europe of the early 1930s that caused so many problems for the US banks might not happen.
It would be interesting to see what effect the loss of the south states have on Government of the people by the people for the people as mentioned in the Gettysburg address.
that this nation, under God, shall have a new birth of freedom—and that government of the people, by the people, for the people, shall not perish from the earth.
https://en.wikipedia.org/wiki/Gettysburg_Address#Text_of_the_Gettysburg_Address
Would it mean and end to Democracy? Would democracy be seen as a less stable form of government compared to monarcy?
Would USA be seen as weak and lead to the great powers in Europe taking more interest in the America north and Latin America?