Postwar relations between Britain and America depends if London decides to make a separate peace. If the British settle, the American public will probably turn the other cheek since the US did start the war in the first place and they have a more important enemy in the south. If Britain backs the Confederacy, the US will never forgive them. The consequences of British support for the Confederacy would include the US losing 1/3 of its territory, its control over the mouth of the Mississippi River, the continuation of slavery in the South, and the loss of revenue from southern crops, an outcome guaranteed to anger almost every Northern voter in some way.
Well, Lincoln is a canny actor and he knows that he has to do everything to keep the Confederacy and CSA from finding common cause. The joint Anglo-Confederate attack on Washington is anxiety inducing, not only as close as it came to succeeding, but that after a year of the two sides treating it as a separate war, the British are willing to openly coordinate with the CSA. His goal in seeking negotiations with Britain is to give them a deal that lets him get on with the original war against the secesh.
If it works, great! If it doesn't or Britain perceives it as weakness...well there's the rub.
I think though, that any situation that sees the war end before the 1864 election would be the one you describe. If the Anglo-Confederate attack on Washington had succeeded, then Britain would truly be forcing the Union to the table, and likely ensuring Confederate independence. The 1864 election however, no matter the outcome of the potential peace talks, is going to be (like OTL) not just a referendum on Lincoln, but the war as well. That makes an enormous psychological difference.
This whole 'the US will never forgive the British' thing seems rather unlikely, in my mind. The PRC forgave the Americans for backing the Kuomintang during the Chinese Civil War. Franco forgave Mexico, France and the Soviet Union for backing the Republicans during the Spanish Civil War. The Soviet Union forgave Japan, the US, UK and Czechoslovakia for backing the Whites during the Russian Civil War. The Irish forgave the British for partition. South Korea forgave the US and Soviet Union for partitioning Korea. Mexico forgave the US for taking over 50% of their territory. Columbia forgave the US for helping Panama to secede. Latin America as a whole forgave the US for the Banana Wars.
Sure, there was lingering enmity in some of these cases, but it didn't stop them from working with each other when necessary. Perpetual hostility to your neighbours is usually just too expensive for nations to afford.
I think it's been pointed out well in this thread that nations don't hold eternal grudges "just because" and often times it comes with other factors. Things got back to normal within a generation of both the Revolutionary War and the War of 1812 (it's important to recall that many people who fought in that war were young or even fought
in the Revolutionary War) and a massive militarized border between the US and British North America wasn't the norm. There's certainly scenarios where eternal hatred of Britain is a thing, and TL-191 does a good one, but I'm certainly not just writing another TL-191!
There's definitely going to be lots and lots of bad blood over the Great American War/North American War/War of 1862/whatever the various sides will call it* whenever it ends. Canadians have now been invaded three times in less than a century and Britain has had to look over its shoulder from Europe at what they perceive as a belligerent American republic. Meanwhile, the United States has, from it's perspective, fought against British interference on three occasions! There will be many people who view Britain as an enemy regardless of the outcome, but others who view it as a "don't poke the bear" lesson from history.
*The war has many names to different people. The South has its own name which is very similar to a certain OTL name that is used.
