What appears to have happened in this case is that Polk rode this slogan to victory, but partly since he knew there was also friction with Mexico (which he was not tamping down), he was in fact willing to compromise with the UK on the Oregon border, and in fact did so after his election.

But as you can no doubt see there was no "special relationship" at this point--just rivalry.

   Nor was there smooth sailing in US-UK relations at the time of the Civil War.   If you want to actually be serious about the topic, you might want to try leaving the pig alone and looking up "Trent Affair", not to mention the "Alabama".

At the time of the "Trent Affair", no doubt there were quite a few in the UK who did not look with disfavor on the prospect that the US would split into two parts permanently--and they would be willing to aid in that endeavor.   Lincoln realized his admininstration would have to back down and did so.

If you read enough history you have a better chance to avoid rash generalizations.