Sure, if the British had cared enough, they could have run the blockade. But eventually there would have been some ships caught. Britain would have been obliged to respond. A state of war would have ensued. Now, as of 1862, the British had fought and lost two wars within 90 years. The war of 1812 was within memory of the British leadership. Within the last ten years they'd fought distant bloody wars in Crimea and in India (putting down the Sepoy Mutiny). There was little desire for another. Further, open warfare with the US would leave Canada in a precarious position, would end US grain shipments to Britain, and might even bring Russia into play as a US ally.
Ultimately, there just wasn't nearly enough upside to Britiain in the deal. There was cotton and there was dividing a rising rival. And that was it.
Should be "fought and lost two wars against the United States within 90 years."