Alot more has to change before the UK recognizes the CSA and declares war on the United States. First the south has to abandon slavery, I think toward the end of the war President Davis offered something to Europe they might have bitten off on had the south not already been all but defeated.
Second, a string of great southern victories will not make the UK jump on board in the war, France maybe. The UK had interest in both sides winning the war, but not really entering the war themselves. There is also no way the UK will join the South after Lincoln issues the Emancipation Proclamation in 1862, it would make the UK look like they are rushing to support slavery. There is also the fact the public of Britain was very pro-Union, but that had a tendency to swing back and forth, after the Trent Affair it was pro-Confederacy, after the EP it was pro-Union where it remained throughout the war.
Thirdly, Britain was very well aware of the costs it would take to fight a war with the United States. Yes Britain had a huge Empire, but it didnt get one by blindly going to war for the hell of it. There were also parts of the Empire that weren't all that loyal and troops were needed to keep insurrection's down. Training an army just to fight in North America, then shipping them there will take some time, not to mention money. There was also no guarantee that Britain would win. The US had the advantage of playing at home, they could turn the war from an offensive war to a defensive one. America was so large, a blockade wouldnt be nearly as effective as the one on the south. The US industry was in its infancy, but still capable, cutting off the supply to the rest of the world could boost the US industry, and with the midwest the United States had plenty of agriculture.
There was also the potential of a Russo-American alliance, Tsar Alexander of Russia was a vocal supporter of the United States in the war, and was licking his lips at the thought of an alliance with the United States, especially against Britain. His main goal was he wanted Constantinople, but the biggest problem there was the British didnt want him to have it. With an alliance with the United States, Britain would have a hard time against the alliance. The United States could take Canada, and Russia could over run Eastern Europe. Britain was all too aware of this proposed alliance, and was none too happy about it, the mere thought bothered them, that was an extremly potent threat, so why would the UK want to give the United States a reason to be willing to accept the offer?
The only way the UK goes to war with the US during the war, is if the US provokes it.