A most interesting query indeed. It seems entirely illogical that we use the term to describe relations between the former colonial master and rebellious (13) colonies. I should however note that this we should not misinterpret the term "special relationship". It is somewhat misleading in that it leads to images of an amicable or even particularly close relationship between the two countries: it does not. The term "Special Relationship" is almost quite the opposite: a relationship which is so special not because it is a close one, but because it's almost unusual in nature.

So with that preamble out of the way, I think we can safely say that such a relationship did not develop with France for a simple reason: by definition, there is nothing special about America and France's geopolitical relations in the past. It was not uncommon for France to assist British colonies in attempting to break away from the rule of the crown or Westminster, but it is uncommon that such a colony then maintain amicable (if at times tense or distrustful) relationships with London for so long afterwards. Let's begin by delving into the sources and origins of this "Special Relationship" then shall we?

The term itself was not actually used at all up until the later years of World War II and early into the Cold War, when Winston Churchill coined the term to describe the need for a new geopolitical relation between the two countries. Here are a duo of excerpts from speeches (dates shown in brackets) where the term was used:

"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).^(1)

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).^(2)

It's clear then that it was Churchill who gave rise to the term "Special Relationship" and clearly breathed into it the majesty and historical grandeur that he so commonly bequeathed to his own "Churchillian" terms. But mind you, 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 (as you shall see in the sources) 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.^(3)

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.

As the Second World War gave way to the Cold War however, many historians (and indeed some American politicians) became skeptical of Churchill and Westminsters newfound appreciation for their geopolitical ties. Here's an excerpt from Truman's Secretary of State Dean Acheson (a legend in Cold War American politics who I highly recommend you look into if interested!) on the relationship:

Of course a unique relation existed between Britain and America—our common language and history ensured that. But unique did not mean affectionate. We had fought England as an enemy as often as we had fought by her side as an ally.^(4)

Throughout the Cold War, the "Special Relationship" has existed in some way or another between the heads of the two nations: Richard Nixon, Harold Wilson, and Margaret Thatcher are just three of the heads of state who made mention of the special relationship in some way or another. But just as where there were those who agreed with the relationship, there were those who dissented from it. British historian Max Beloff saw it as "an agreeable British 'myth' to help cushion the shock of national decline",^(5) whilst Acheson (in similar rhetoric to his previously cited comment) viewed it as "a dangerous intellectual obstacle to acceptance of Britain's largely European role."^(6)

Still curious about the "mythos" of a relationship extending prior to World War II? Here's a quote from historian Ray Raymond proposing a rather interesting answer to why the two nations are able to get along so well despite the usual (and oftentimes problematic) diplomatic setbacks:

That the solution to the mystery, the reason why the relationship is special, is that so much of the basic DNA of the infrastructure of the American political, legal, and economic system is British... In a very real sense, therefore, the United States, however foreign it may sometimes appear to many modern-day Britons, is - to borrow David Hackett Fisher's memorable phrase, - "Albion's Seed".^(7)

So there we have it then. A relationship whose origins, nature, and evolution remains a constant topic of discussion for both historians and geopolitical analysts alike. A relationship that was not quite (as the question initially poses) the natural byproduct of friend or foe on the fields of the Revolutionary war. A relationship whose "special" quality has been tested and called upon by the occupants of Number 10 Downing Street and 1600 Pennsylvania Avenue more frequently in the past century, even if it took the "British Bulldog" Churchill to give it its grandiose name.

Hope this helped shed a light on your query!

Sources and Further Reading:

[1]: Quoted and Cited in Brewer, Susan A. To Win the Peace: British Propaganda in the United States during World War II. ITHACA; LONDON: Cornell University Press, 1997. Accessed November 13, 2020. https://www.jstor.org/stable/10.7591/j.ctvv41253.

[2]: Transcript of Winston Churchill's "Sinews of Peace" Speech, delivered March 5, 1946. Accessed November 13, 2020. https://winstonchurchill.org/resources/speeches/1946-1963-elder-statesman/the-sinews-of-peace/

[3]: Reynolds, David (April 1990). "1940: Fulcrum of the Twentieth Century?". International Affairs. 66 (2): 325–350. JSTOR 2621337

[4]: Acheson, Dean (1969). Present at the Creation: My Years in the State Department. New York: W. W. Norton. p. 387. Accessible here: https://archive.org/details/presentatcreatio00ache.

[5]: Reynolds, David. "A 'Special Relationship'? America, Britain and the International Order Since the Second World War." International Affairs (Royal Institute of International Affairs 1944-) 62, no. 1 (1985): 1-20. Accessed November 13, 2020. https://www.jstor.org/stable/2618063?.

[6]: Reynold, David. "A 'Special Relationship'?"

[7]: Raymond, Ray. U.S.-UK RELATIONS AT THE START OF THE 21st CENTURY. Report. Edited by McCausland Jeffrey D. and Stuart Douglas T. Strategic Studies Institute, US Army War College, 2006. 1-14. Accessed November 13, 2020. http://www.jstor.org/stable/resrep12126.5.

The special relationship as we think of it today really emerged in the 1890s, prior to this US-British relations were rocky although not hostile. After the War of 1812 a few truths were widely being noted. First, was that US-British trade was economically important for both countries, originally for New England, but later with the spread of cotton for the south. There were high economic costs to engaging in conflict, more so for the US than for Britain, but going to war for no reason wasn't going to be popular with a lot of economic interests on either side of the pond. British imperial ambitions in the 19th century had moved away form the Americas. They still held a long-standing dispute over British ambitions in Central America, but the issue wasn't tense enough for either side to go to war over. Britain was also in a situation where it's ability to project power was limited. In the early part of the 19th century there was a realization that Britain could fairly easily blockade the US and defend its possessions but probably didn't have the ability or the will to win a decisive victory. In both the Revolution and the War of 1812, the major British efforts to invade American soil failed sometimes with disastrous consequences. After the Civil War it became fairly clear that the the British would have difficulty defending Canada in the case of war and if the Americans took it, the British would be hard pressed to gain it back. Agricultural exports from America and Canada were important and if other European great powers took the opportunity to fight Britain (especially Russia and later Germany) while Britain was hard-pressed in America it could very well over extend itself and lose the war. So Britain had a lot of incentives especially once the Germans began a naval race with Britain, to keep America neutral.

The major issue of British-American relations in the early 1800s was westward expansion. Even after the War of 1812, the US wanted to prevent America expanding too much into the interior, and had ambitions on the pacific coast of their own. There was also various border disputes between the US and Canada and the problem of a virulently anti-British Irish-American minority launching raids into Canada. Tensions over this briefly flared in the 1830s, but both sides de-escalated things and built a degree of trust between the parties. The US started arresting people trying to aid the Canadian revolt, they settled their dispute over Maine's borders diplomatically, and the British while providing some aid to Mexico and floating the idea of preventing the US annexation to Texas eventually dropped it. While tensions would flare again over Oregon, Nicaragua, and more seriously over British aid to the Confederacy, after the crisis of the late 1830s, the US and Britain generally had fairly normal relations and neither side felt exceptionally threatened by the other. The civil war briefly spiked tensions on both sides, but once the British were willing to pay for damages caused by the British built CSS Alabama, this apology was mostly accepted by the US. I wouldn't say in the 1870s that the US and Britain would consider themselves close allies, but they didn't see each other as adversaries either, although American politicians occasionally engaged in anti-British rhetoric to win Irish votes in cities.

The real turning point came in the 1890s. In an increasingly diverse America a lot of the old anglo-americans adopted an Anglo-Saxon ethno-nationalistic identity which, saw Britain as "real" America's cultural and ethnic sister state. Many elite prep schools and colleges started teaching this in what is called, "the imperial school." There was also a brief spate of tension with the US in the Venezuela Crisis and the Alaska border dispute and the US vigorously enforced its Monroe Doctrine, but in a diplomatic way.

Britain came out of this with a few realizations

1. The US had a generally friendly attitude with Britain and as long as Britain staid out of the Americas it could have good relations with the US
2. The US was a great power and powerful enough that the British couldn't bully it, the US was also not really stepping on Britain's toes for its imperial expansion elsewhere so if the US wanted a dominant role in the Caribbean that was fine. Some of course disagreed
3. Tensions were beginning to rise between Britain and Germany, and having the major power of the Americas as an ally rather than an enemy was helpful

After that Britain and the US enjoyed good relations until the beginning of the depression, in which economic nationalism briefly created some bitter trade disputes. This relationship was repaired though and the US and Britain have continued to feel like they are closer than other countries as perhaps a byproduct of long good relations and the after effects of Anglo-Saxon nationalism that was popular in schools during the late 19th and early 20th century. They also have tended to have a coincidence in interests in foreign policy matters which has further helped the relationship

Thank you for those two detailed and informative answers. Be great to hear a little bit about the French side of it all, i.e. why didn't France's role in supporting independence lead to close French-US relations?