Did anyone in the world other than Britain, the US and possibly France care?
Spain did.
Can you elaborate a bit? It would seem Spain would just sit back and watch as two rivals beat the crap out of each other. True they had Florida right there, but I would think they wouldn’t be concerned about their western possessions since they were a 100s of miles away and America was pretty weak after the revolution.
It set a precedence for independence from the European home country for other colonies. Spanish colonies, in particular, could point to the US and say “If those English colonies can become independent, then so can we.”, although they weren’t the only ones (Haiti, also). In addition, some of the liberation rhetoric espoused during the American War of Independence was influential in the French Revolution.
ETA after seeing the above post: Spain eventually joined in the American War of Independence on France/American side. And Florida, at the time of the war, was actually a British possession. I believe Spain reacquired it as a result of the war, although I’m not 100% sure of this.
If I recall, Spain sent money & guns, & some of their fleet, late in the fight.
Yes, they did
from wiki–
All the European countries with colonies certainly cared that a colony revolted, fought a war, and successfully seceded from the home country. Moreover, the American situation was an offshoot of what was essentially a world war between France and Britain that dragged in other countries from time to time. Everybody in Europe cared about that, as long as everybody is understood as those upper classes (and members of other classes like merchants and scholars) who looked beyond their borders. And proxy wars elsewhere often lead to hot wars on home turf.
In the same sense, but somewhat wider, monarchies had spent the century being unsettled by growing protests among their peoples. Any successful action against a monarchy was threatening to them.
As an individual immediate event, the French Revolution was the Big Deal. But few in the upper classes In Europe could fail to make the connection between the two.
Did anyone outside Europe care? Probably not, except as they cared about other events happening in Europe.
Did anyone else care at all? Of course. Did anyone care all that much? No. Heck, even Britain didn’t care all that much. If they had put the full resources of the British Empire into fighting the insurgent colonists, as if they were fighting for Britain’s survival, they would very probably have won. However, they had other, bigger and more urgent priorities, and did not do that.
Intellectuals all over Europe cared to the extent that the United States were the first system of government built upon the ideals of Enlightenment, which was pretty much en vogue among political philosophers at the time. In that sense, it was probably seen as something of an experiment, of putting into practice something that European philosophers only theorised about.
Russia quite a bit at least among the aristocracy because of the market here for hemp in shipbuilding and various other trades. And partly from various involvements in the previous war/wars.
One of the most well-known and admired people in the world at the time was Benjamin Franklin. I think his stature as a scientist, author, philosopher,etc caused people, especially in the intellectual circles, to take notice of the American Revolution.
Not yet it wasn’t.
I get the impression British schoolchildren are taught some version of “Aww, we didn’t really want those nasty old colonies anyway. Nobody cared what they did.”
The American Revolution in general and the Declaration of Independence in particular were very influential on intellectual and political thought in Europe and Latin America in the 18th and 19th centuries. From the wiki article on the Declaration of Independence:
The American Revolution also influenced Irish dissidents and Italian nationalists.
I think a lot of the intellectual influence that people are talking about followed more from the subsequent success of the USA as a nation (particularly after the adoption of its constitution) rather than from the revolution/war of independence itself. If the USA had degenerated into a poverty stricken mess within a few decades of gaining its independence, or had remained just a few sparsely populated statelets perched on the edge of the continent, with vast Indian (or French, or whatever) controlled lands behind them, no-one would have been much impressed by the fine words of the Declaration of Independence or the intellectual credentials of some of the rebellion’s leaders.
But, apart from that, of course people elsewhere took some notice of the conflict between the colonists and the British, and its outcome. No doubt, if you lived somewhere where there were newspapers that reported international news (a fairly sizable if), it would have been covered in the newspapers. The OP’s question, however, was “Was it a big deal?” Well, big compared to what? Compared to the French Revolution that broke out a few years later (admittedly partly, but only in quite small part, inspired by the American one), or to the Napoleonic wars that followed on that, or compared to many earlier European wars, or the English Civil War, which was not all that long past, no it was not a big deal for most Europeans (and even less for, say, the Chinese), and not a terribly huge deal even for the British. It was a sideshow.
Of course, in hindsight, now that America has evolved into the world’s dominant superpower (and probably even well before that, by the mid 19th century, say), the American Revolution could be seen by anyone to have been a crucial turning point in world history, but at the time, not so much.
Inasmuch as they are taught about it at all (which isn’t, generally, very much), British schoolchildren are likely to be given the impression (though probably implicitly rather than explicitly) that the outcome of the American War of Independence was “a good thing”, that it was right and proper that America became independent. They won’t be taught that the British were the villains of the piece, but neither will they be taught that the outcome was a great disaster for Britain.
Indeed, it was not a great disaster for Britain, whose empire, prosperity, and economic and military power, continued to grow, and to eclipse its rivals, for more than a century following the loss of the North American colonies. (The real disasters for Britain, that eventually came, were World War I and World War II.)
Australia certainly didn’t. 
You mean because the Declaration of Independence was 1776 and the Constitution 1787? Well, firstly, there’s a lot of talk about government based on Enlightenment ideas in the DoI as well, Secondly, even the Constitution was before the French Revolution, when political ideas from Enlightenment were first tried on a large scale in Europe.
Technical nitpick: Britain didn’t lose “the North American colonies,” it lost MOST of the North American Colonies. Quebec, Newfoundland, Jamaica, etc., stayed British for a long, long time.
Edit: Wiki has a nice list.
It wasn’t a big deal at the time, outside of the northeast.
Most everyone else was busy hunting, fishing and texting.
See my post #13. The U.S. constitution certainly had not had time to prove itself a success before the French revolution broke out. Also, of course, most of those Enlightenment ideas that inspired America’s founding fathers came from France in the first place.
I think it’s worth mentioning that all the Indian nations anywhere near the 13 colonies cared.