First, let me state that boys called "Siri" and "Remi" in the U.S. would have their lunch money stolen every day of their lives. Including adulthood. This is not an American phenomenon, let alone one that is "too American." I think it's just too "teenage girl," in which Sirius and Remus are made into honorary teenage girls. (I don't dislike teenage girls. But, well... let's face it on the name thing; we all go through our "cutesy" stage.)
But nicknames in general? Even before Narcissa Malfoy was canonically called "Cissy," it wasn't "un-British" (or, well, "too American") to use the nickname. I mean, let's have a look, shall we?
The late Princess of Wales was referred to repeatedly as "Di."
Her elder son, at least as of a few years ago, appeared to go casually as "Wills."
Prince Henry Charles Albert David (according to the official site of the royal family) has been called "Prince Harry" since birth.
Joanne Rowling, last I knew, went by "Jo."
The woman born Margaret Natalie Smith (according to IMDB) is credited on film as "Dame Maggie Smith."
Catherine Elise Blanchett was born in Australia (not America) and works as Cate Blanchett.
That's from about ten minutes of interrupted web searching, and just people I could think of offhand to check. Names like "Liam," "Nancy," etc, began their lives as nicknames for other names, and acquired proper status from repeated use... and not on this side of the Pond. So no more of this "Nicknames are too American" business, thanks. There may be particular nicknames that are American (does anyone else call "Henry"s "Hank"?), but the concept of nicknames based on names? So not.
In HP canon
It's quite true that I think Minerva McGonagall would hex anyone calling her "Minnie" into next year, but that's because she's McGonagall. There isn't really a good nickname based on a name for Sirius or Remus (which probably explains why their nicknames are Padfoot and Moony), and Severus has any number of curses scribbled in the margins of his textbooks which would fly at anyone attempting to call him "Sev."
On the other hand...
Ginevra Weasley is always "Ginny," and her brother Ronald is "Ron." We don't know whether Bill is a William or a Bilius, but he's most likely not just a "Bill" (nor is he sitting there on Capitol Hill, for that matter), and "Charlie" is generally a nickname for "Charles."
"Ernie" Macmillan, in keeping with his family's distinguished heritage, is probably an "Ernest" on his legal papers.
"Davey" Gudgeon almost lost an eye to the Whomping Willow.
"Bella" was given her nickname in the same book as her full name, Bellatrix, was given.
Dudley is given numerous nicknames from his mother, but unsurprisingly prefers his friends' version, "Big D."
In other words, like most of life, it depends on the person... but it's certainly not an unheard of practice in the milieu.
I'm sure that the backlash is against horrible nicknames--the "Won-Won" school of nicknaming, which I hope was not reciprocated by calling poor Lavender "Lav." "Siri" is lovely if you happen to be a female Jedi knight in love with Obi-Wan Kenobi in a Jude Watson series, but atrocious if you're the eldest son of the House of Black. Remi... just doesn't really work for anyone. Peter Pettigrew may have been a "Pete" to some people, but I'm guessing from the way his ex-friends refer to him that he was always either Peter or Wormtail, just as James was always James or Prongs... though his parents may have called him something else until he was old enough to object. As to Lily... no evidence either way. I can't see any special reason to call her "Lils," but is that really so awful?
At any rate, it's not an American vs. non-American issue. That's all I wanted to point out.