eventually? Fairly quickly. When calculating the strength of the USN, you have to leave out the monitors and the other ironclads that could only operate on the rivers... and that doesn't leave all that much when compared to the RN. As for the blockade... it'll pretty much ruin the north... along with the sheer economic disaster it would have caused, it would have hampered military efforts too; the north had to import a lot of the stuff needed to make gunpowder; hell, for the first couple years of the war, they had to import guns as well. On the flipside of it, the CSA would suddenly be open to imports again, and would be able to get everything they need from overseas. Not to mention, the financing of the Federal government was somewhat reliant on customs fees, and the blockade would end all that....
Except history is littered with nations fighting long past economic disaster .. one example being the actual CSA in RL, but we can throw in Paraguay, Japan, Germany (twice), etc. We are past the first couple of years of the war and the US has fully mobilized its economy and has a lot of slack (historically its growth was tremendous during the war, and had enough slack to settle Colorado, build up an extensive infrastructure west of the Mississippi in Kansas, Minnasota and parts of Missouri). Plus this is going to be a ruinously expensive war for the British too (which was a real life concern for them at the time and probably why there was never a historical third war between the US and UK).
However, the blockade would be at least partially lifted for the CSA. They would be able to sell cotton in the UK. Possibly the British government would allow loans to the CSA government. But the strategic situation for the CSA is pretty grim in the fall of 1863. The Mississippi River is under complete Union control, northern Virginia has been thoroughly wrecked economically in the last two years, and the South has suffered tremendous losses and does not have the high quality army it used to have. Many of its best are dead or crippled (army, corps, division and brigade commanders), while attrition (and firing a lot of incompetents) has actually made the US Army stronger.
The best the CSA can hope for realistically is to hold off further attacks and hope the British can defeat the Yankees before attrition bleeds them out. Which is not good news for the British, as attrition politically is potentially very dangerous for the British government. Its the main reason they were willing to accept status quo ante bellum to end the War of 1812 and accepted defeat in the War of Independence to begin with.
Basically what we are saying is the British cannot win quickly. Not that they can't win possibly (perhaps probably). But the US is not in a hopeless situation. Particularly as fighting the Russians is going to require sizeable (perhaps larger in numbers) of British troops and ships as well. The French may or may not be of much help, as a the French have sizeable commitments in Mexico, don't care really who wins the Civil War as long as the US can't kick them out of Mexico, and I can't imagine the French providing serious assistance to help the British in Canada or in the Pacific. Possibly the French will help against the Russians.
Napoleon III was not the most competent or wisest leader the French ever had so its hard to say.
So if Lincoln can retain political support during the 1864 elections (a year away) the US will stay in the war. Based on the setup its pretty clear the British are the bad guys (in the eyes of Americans) in this situation, so a foreign war with the British is far more popular then the situation up to now. Plus the CSA is siding with a foreign aggressor, and lets be clear, there is no compelling reason of national survival or even necessity for the British to be fighting this war. So argueably, the CSA are now a bunch of traitors (even more then they were before), helping foreign enemies invade the US (which has happened in the book). Remember, the British start the war by fighting a naval battle practically in New York Harbor, then attack Portland without warning and shell the town.
As set up, I think historical American reactions to such attacks speak for themselves. Those kind of things unite Americans to the point where we will even let our Presidents invade countries that argueably had nothing to do with the attack on us to begin with.
The British cannot conquer the US, they can only inflict devastating (at times) blows. The US can also deliver some pretty severe blows to Canada (eventually.. remember in the book there is a sizeable army assembling in Detroit).
How long with the British be willing to fight a war like this when the political and economic costs are going to potentially outweigh the gain.
Such costs include making a permanent enemy of the US if the South is able to secede and eventually the US will be able to crush the South (especially a South which has lost Texas and Louisiana, which seems to be alluded to in the first book).
Some of the dissenters to this position are focused too much on hardware and numbers and not enough of how wars are fought and why they are lost and won. Its not how much force and power you have, its whether you can apply that force and power to force a decision.
I for one do not think that is clear cut in the least.
