Hm.. that rather ignores a couple decades or more of a major social political & cultural turbulence in US history. It ignores a few bits of anti slavery activity & policy in Britain as well.
It doesn't ignore the expressed foreign policy of the Lincoln government, though.
'you will not consent to draw into debate before the British government any opposing moral principles, which may be supposed to lie at the foundation of the controversy between those (the Confederate) States and the Federal Union' (Seward to Adams, 10 April 1861)
'refrain from any observation whatever concerning the morality or immorality, the economy or the waste, the social or the unsocial aspects of slavery... the condition of slavery in the United States will remain the same whether [the revolution] shall succeed or fail... the new President, as well as the citizens through whose suffrages he has come into the administration, has always repudiated all designs whatever and wherever imputed to him and them of disturbing the system of slavery as it is existing under the Constitution and laws. The case, however, would not be fully presented if I were to omit to say that any such effort on his part would be unconstitutional, and all his actions in that direction would be prevented by the judicial authority, even though they were assented to by Congress and the people.' (Seward to Dayton, 22 April 1861)
It doesn't ignore the views of the border states, either.
'The undersigned, Representatives of Kentucky, Virginia, Missouri, Tennessee, Delaware, and Maryland in the two houses of Congress, have listened to your address... The right to hold slaves is a right appertaining to all the States of this Union. They have the right to cherish or abolish the institution as their tastes or their interests may prompt, and no one is authorized to question the right, or limit its enjoyment. And no one has more clearly affirmed that right than you have... satisfy them [Southern moderates] that no harm is intended to them, and their institutions: that this government is not making war on their rights of property, but is simply defending its legitimate authority, and they will gladly return to their allegiance... Confine yourself to your constitutional authority: confine your subordinates within the same limits; conduct this war solely for the purpose of restoring the constitution to its legitimate authority; concede to each state and its loyal citizens, their just rights, and we are wedded to you by indissoluble ties'.
Britain might have exerted more diplomatic pressure for the Union, but this would have had negligible effect on the war.
In the highly unlikely scenario that the Union government goes all out for anti-slavery, the British might have been a bit more favourable. However, that's completely outbalanced by the loss of sympathy in the Border States, and indeed in the large section of the Northern population more generally who were prepared to fight for the Union but not for the slave. That's why Seward gave the instructions he did to his ambassadors, why Lincoln cancelled moves toward emancipation by Fremont and Hunter, and why as late as 1864 he was prepared to write - although not to send - a letter saying 'if Jefferson Davis wishes, for himself, or for the benefit of his friends at the North, to know what I would do if he were to offer peace and re-union, saying nothing about slavery, let him try me.'