Greetings! This is certainly an interesting question with regards to the origins and nature of international relations, and is actually a rather pertinent question in the post Cold War era nowadays as well. There are however, some serious misconceptions with Anglo-French-American relations that OP has put forward in their explanation of their question, so we must turn to those first before understanding firstly why the "Special Relationship" was not actually as Special in its origins as some mainstream sources have put it, and secondly why the "longstanding rivalry" between Great Britain and France continued to persist even when they were allies. Let's begin with clearing the air around this topic area:

Note: As my area of research and body of knowledge is mainly to do with the British perspective of international relations, any other comments on the American or French view of the other two nations would also be highly valuable.

Misconceptions

"the English and French have been fighting each other for so long you might think it was the official pass time for them all to meet up in Normandy and pound the tar out of each other."

Whilst an amusing note, this ignores the fact that from 1815 onwards, the British and French actually reached a rapport of sorts between their two countries. Granted, the two nations remained at loggerheads on issues of colonial territories, most notably over the British invasion of Egypt in 1882 and its "administration" of the nation, and the Fashoda Incident of 1898 (more on that later). However, the 19th century had also seen the rise of Anglo-French cooperation on more matters as well. OP has already noted that both countries were in rough economic support (though loose and never officially endorsed) of the Confederacy during the American Civil War. It should also be noted that the British and French fought alongside each other during the Crimean War of 1853 to 1856, and their expeditionary troops also collaborated as far afield as the Qing Empire during the Second Opium War of 1856 to 1860.

Granted, the British government at the time certainly maintained an air of caution when dealing with the French (and vice-versa), but by the turn of the 20th century it had become clear that both powers required the support of each other, and that their longstanding rivalry would need to be put aside to face the greater threat (at least, the greater perceived threat): the German Empire. This previous response deals with the rise of the German Empire, or Kaiserreich, as a key motivating factor for the Anglo-French rapprochement in the Entente Cordiale of 1904. For a condensed yet insightful version of that discussion however, John Darwin (whose work on the British Empire from the 19th to 20th century I highly recommend) has the following on the German threat to London.

"The novelty and seriousness of Germany's naval challenge - and the reason why it aroused such a fierce reaction in Britain - was that it threatened to nullify the British claim to be a great power in Europe, a claim founded ultimately on the possession of sea power."

Thus to address the first misconception as quoted at the beginning of this section, the 19th century had seen several "sores" in Anglo-French relations, but it was by no means surprising (or indeed, extraordinary) when the two nations drew closer together to deter the mutual threat of an ascendant Germany in 1904; though not, as it must be stressed, in a mutual defense pact, but rather a stronger "informal alliance". By the beginning of the First World War, the Anglo-French governments had already put aside their rivalry stretching back centuries, and had entered (since the mid 1800s) into a more amicable (if at times suspicious and strained) state of relations. The sentiment below, delivered by British Home Secretary Lord Palmerston during the prime ministership of Lord Aberdeen (1852 - 1855), was not shared by his successors of the early 20th century:

“[France’s greatest goal] was the humbling of England, the traditional Rival of France, and the main obstacle to French supremacy in Europe and all over the world.”

A Not-So-Special Relationship

Turning then to the second misconception in the explanation of the question, we have this bit before us:

"Then we get to World War 1 and suddenly the three of us are all the bestest friends you ever heard of, unquestioningly having each others back basically up until and including today."

The expression "bestest friends" is a tad misleading in that it paints a picture of Anglo-French-American cooperation and collaboration from the First World War which was filled with complete loyalty and trust in each other: this is false. Even during the First World War, President Wilson viewed "British navalism" to be just as bad as "German militarism", and many within his cabinet supported American neutrality up until 1917. In this response, I go more in-depth as to why America joined World War 1, and that it was clearly not just a matter of "supporting Britain and France" for the sake of being a good Allied power with them.

Even after the First World War, the British and French delegations at the Paris Peace Conference viewed Wilsonian idealism as "misplaced" and "detached" from the current world situation (or, more accurately, their own visions of the post-war order). In fact the very phrase "Special Relationship" has Churchillian origins, as exemplified in these excerpts from two his speeches:

"It is my deepest conviction that unless Britain and the United States are joined together in a special relationship... another destructive war will come to pass" (February 16, 1944, to British politician Richard Law).

Neither the sure prevention of war, nor the continuous rise of world organization will be gained without what I have called the fraternal association of the English-speaking peoples ... a special relationship between the British Commonwealth and Empire and the United States. (March 5, 1946, delivered in his famous "Sinews of Peace" Speech at Fulton, Missouri).

Thus, the fact that a British Prime Minister coined and used the term calls into doubt its authenticity: did the American Presidents across the pond see the relationship with their former colonial overlords the same way? The simple answer to that question: not quite.

Historian David Reynolds, who is something of an expert on this topic and has written a number of articles and books on the matter, portrays the pre-WW2 nature of Anglo-American relationships in a rather bleak light:

For most of the period since 1919, Anglo-American relations had been cool and often suspicious. United States 'betrayal' of the League of Nations was only the first in a series of US actions — over war debts, naval rivalry, the 1931–2 Manchurian crisis and the Depression— that convinced British leaders that the United States could not be relied on.

We can probably then, ascribe the Special Relationship as a product of wartime necessity rather than outright diplomatic nicety or historical precedence. The Second World War ushered in the need for Britain and the United States to draw closer as the key Western Allies and increase the level of military, political, and economic cooperation to an amount hitherto unseen. Churchill and Roosevelt did so, and their relationship is seen as one of the more prominent examples of Anglo-American cooperation and to a large extent, the embodiment of the "Special Relationship" which was needed to defeat the Nazi threat.

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