To start, I'd like to point out that essentially, this is an expanded-upon review I wrote on the piece Comparative Study by fringeperson. It is essentially an essay opener on how the HP world is presented against some popular American magical examples, and how these worlds would coexist, with some other parts, such as Asian magical ideas and Mary Poppins, which are considered in brief as well. I would like to note that the original piece is effectively an essay opener, because a lot of avenues are not explored.
I like the general idea of multiple magical worlds coexisting, but the trouble with this is that each title possesses very different magic. Magic which a lot of people don't seem to understand wouldn't be able to coexist in the same universe under their individual rules without overlapping ridiculously, and therefore disallowing them to coexist as suggested. The simple point is that in any crossover, some things from one world must be more or less useful than an equivalent from the other, because two different worlds with different rules wouldn't allow things to work exactly the same but with different methods of delivery. Most fanfiction writers certainly will not, at any rate.
The trouble with the HP universe in terms of spell-casting is because its based around the ideal of the magical staff, a centrepiece for magic in Europe and specifically in the legends of Arthur, from which Merlin appears. Merlin obviously is considered the cultural icon of magic in Britain, therefore anything related to him would no doubt be included in British works. The wand is merely a more compact staff, and has been considered thus for centuries in mythology. Druids and Merlin are depicted with staffs, as is Odin in the Norse legends. Gandalf, essentially derived from the legends of Odin, also uses a staff. Since these are the most well-known European figures, it is understandable that the wand is the tool of trade for the HP wizard.
Meanwhile, I'd posit that the idea of western spellcasting without an instrument or tool is due to the cultural spread of Wicca and other magical religions since the 60s, which is something much more popular in the States than in Britain. Specifically, Charmed has a definite basis in Wicca, though it has also been criticised by people who call themselves Traditional Wicca, similar to how there are Protestant and Catholic Christians. The differences between Asian cultures and European cultures for magic is again based in religious viewpoints and mythology, but as I do not have much if any knowledge in their legends, I won't offend them by not doing them justice.
With Mary Poppins, it's fairly obvious that she couldn't exist outside of a fictional setting, simply because her characterisation includes the following: "She is practically perfect in every way." This implies she is the absolute perfect state for humanity, and thus she would the subject of vast amounts of envy and hate because such a person cannot exist, by and large. In fact, she appears to be a personification of the Christian God, and has been eluded to being as such in at least one publication. So she's essentially a cheatsheet used by people trying to justify why wands are stupid, because Poppins is British and doesn't use one.
Onto the second-last point of the piece: What if Harry Potter was American rather than British? Which to me is a two-part question - what would his magic casting be like if he were American; and would the setting of America be better or worse than the current setting.
To me, it's very obvious that if HP was American rather than British, he'd still have to use a wand. The simple fact is that the HP universe says that wandless magic is incredibly complicated and difficult. We know this from simple facts, such as the fact that a powerful wizard such as Voldemort, someone who knows that wandless magic is possible, still uses a wand. If it could be considered such a huge weakness as fanfiction authors make it out to be, why would a genius like Tom Riddle Jr. use one?
The only obvious answer, if we ignore that JKR never really considered that it would be questioned, is that wand magic is much more powerful than that of wandless magic. You cannot argue that Voldemort was a genius in school, because he is described as practically the perfect student - intelligent, charming and apparently able to hide his distastes from everyone who he chose to. Therefore - it is obviously infinitesimally easier to get results of a similar magnitude with a wand.
Plus, this ideal that America would make a much better setting than Britain doesn't seem very well thought-out. In Britain, it is still quite possible, though rarer now, to find a boarding school for the gifted or the problematic for children the age of the students in HP. America is much more about the high school, which while being the subject of every other children's TV show, doesn't truly mix with this idea.
Besides, for all the mocking of the traditional ways of the Wizarding world, it would make sense to consider that perhaps the long-standing ways of said world would have more knowledge than the American equivalent. In fact, the idea that Muggleborn would be better off in America is very amusing - America is a country which is very right-wing. The Muggleborn equation, from what I understand of it, would actually be something much more socialist and left-wing.
After all, why should the aristocratic Purebloods, whose families may or may not have had to work extremely hard to make said money, allow some uncultured philistines to gain a foothold in the Purebloods society? Muggleborn who do not appear to appreciate the cultural values of the Purebloods and damn them for wishing to remain Traditional in that sense. To me, undercutting the rich and pushing up the ignorant is very socialist, something more likely in the UK than the US.
Plus, the War of Independence was fought by the rich and aristocratic living in the US. Surely if the Wizarding World of America also seceded from Britain, then it would most likely be even more Pureblood dominant than the UKs - the Purebloods have all the money, and in a right-wing state like America, he who has money has a right to do what he wishes with it. So I doubt that would change somehow.
The final point is: what if Hermione Granger noticed the difference and wrote a comparative study?
Well, who is to say Hermione Granger is the most intelligent witch of her generation, and therefore the first person to notice a difference? This is how I interpreted that question - because why specifically Hermione if not for that very reason.
As far as we are aware, Sirius is the only one to really make the point, but he says that he's ever met. Because an escaped criminal has blatantly had the opportunity to meet every witch of that generation in every country in the course of... what, a year tops? Other than that, only the lead characters appear to agree with this notion, but I'd argue that the most intelligent females of Hermione's generation who could be considered main characters are either Tonks or Fleur.
Tonks is an Auror, which is made clear to the reader to be a highly sort-after career and needing an extremely good education to achieve. Meanwhile, Fleur works part-time at Gringotts. She has managed to achieve a job in what is apparently a very Goblin-dominated firm or company, the year after she has graduated. At most, she would be 18, according to the HP wikipedia. The only reasons to say Hermione is the most intelligent is either because a) she's the main female character and therefore the only one we really explore; b) because someone thinks that Emma Watson was the best actress; or c) because Emma Watson is hot. Hardly good reasons to consider Hermione the most intelligent.
Honestly, this is merely just my attitude to the very blasé attitude of "America yay! Britain boo!" to the typical crossover between two magical stories. A lot of stories which do their best to explain reasoning for this are among my favourites that I've read, but that's the key difference. "Oh, it's Japanese/ Chinese/ American/ insert cultural idea here so it's better than the wanded magic of Britain." That is, quite frankly ridiculous, because of how the HP universe works. "This is better in this culture compared to the wanded art of x, because of reasons a through to g, but this wanded art is better than the inverse because of reasons h through to n" - that's the sort of thing that is honestly needed if you feel the urge to compare something with HP.