I would say that little actually changed between the Civil War and WW1- rather, the crucial era of forging the 'special relationship' was the 1930s and 40s.

Relations between the US and UK remained shaky in the 50 years following the Civil War. Indeed, the two were brought to the brink of war in 1895 over disputed territory in Venezuela (covered by America's Monroe Doctrine assuring hemispheric predominance) and British Guiana (a British colony). This continued well into the period of the First World War- it's worth remembering that Britain had been at war with Germany for two and a half years before the US joined the conflict.

Even during the war America held itself at arm's length from the imperial powers. It refused to classify itself as an official ally and instead fought Germany as a co-belligerent. This is why the official name of the allied coalition was the "Allied and Associated Powers", with the USA firmly in the latter camp.

These divisions were exacerbated after the war when the USA declined to join the League of Nations (the brainchild of its own President Woodrow Wilson) and sheltered behind restrictive tariffs which fostered resentment on the part of a free-trading UK. The economic catastrophe wrought by the collapse of Wall Street only furthered anger; President Roosevelt would go on to collapse the 1933 London Economic Conference designed to revive international trade after denouncing British currency stabilisation.

There were even war plans considered `between the two in this period (although the likelihood of their being carried out was remote). If you fancy some interesting further reading, 'Basic War Plan Red' was the American plan to conduct military operations against the British Empire.

All this considered, however, it is clear that the US and UK shared some key characteristics- language, culture and (imperfectly) democratic institutions. These common values were thrown into sharper and sharper relief as the threat from fascist and communist challengers grew in the 1930s. By 1940, both Britain and the US had leaders (Churchill and Roosevelt) who recognised the need for both nations to work together to defend their common interests against the Nazi threat.

It was at this point that a special relationship began to grow- there was precious little evidence of it in practice beforehand.