I thought it would never happen, but it did. Lynne Murphy has alerted me that the American site Slate has published the following sentence. (It’s from the Dear Prudence advice column; a woman has written in complaining that all her friends want to talk about is their darling baby grandchildren.)
Let’s hope these two can get free from the nappies long enough to come to brunch at your house or join you somewhere for coffee.
There is no reason for no reason for an American writing for an American publication to use “nappies” instead of “diapers”–other than being cute, that is. There’s nothing wrong with being cute, of course, but I would be gobsmacked if “nappies” caught on here.
notoneoffbritishisms.com 
I wouldn’t bet against the cuteness factor — or discount the influence of other trendy “-ie” words like selfie, lippie (lipstick), and sunnies (sunglasses).
The ‘ie’ ending is a common Australasianism. When N Americans would call truck drivers ‘truckers’, Ozzies and Kiwis call them ‘truckies’ and when N Americans would call ‘outlaw’ (‘one percenter’) motorcycle club members ‘bikers’, Ozzies and Kiwis call them ‘bikies’.
A smoking break from work is a ‘smokie’, a sick day is a ‘sickie’ and a bathing costume (swimsuit) is a ‘cozzie’. ‘Sunnies’ for sunglasses is certainly common in Australasia.
And so it goes.
Sometimes we use “‘o” endings, smoko not smokie, for instance or ‘garbo’. The rest cited are spot on. cf, ‘firies’ ‘tradies’, ‘pollies’ for firefighters, tradespersons and politicians.
My husband and I watched the Aussie drama Rescue: Special Ops in UK. We would gleefully shout ‘drink’ at every -ies or -os abbreviation (6pm on a week night being too early for actual drinking games).
Most have been mentioned so I’ll add ambos and Salvos. I’m sure there was one for fisherman but I can’t remember it.
Cf.
It was a Briticism that got some US press due to Prince George a while ago. (Like this Today Show bit. So, like Nancy, I’m not hugely surprised. I think synonym-avoidance has something to do with it in this case too.
When boxing up books for a library (in the US) a few years back, I encountered a packing box of Huggies (or one of their competitors) that was labeled:
Diapers — Couches
Pañales — Nappies
UK has always had secondary meaning of diaper, but nappy < napkin seems to have squeezed it out: OED-
a1616 Shakespeare Taming of Shrew (1623) Induct. i. 55 Let one attend him with a siluer Bason Full of Rose-water, and bestrew'd with Flowers, Another beare the Ewer: the third a Diaper.
1837 H. Martineau Society in Amer. II. 245 Table and bed~linen, diapers, blankets.
1879 J. M. Duncan Clin. Lect. Dis. Women iii. 31 You cannot judge of these discharges when dried on a diaper.
I recall a colleague at work saying that a haughty friend of his mother heard an American woman say “diaper” to which she retorted, “We stopped using that word centuries ago.”
May it not be possible, that apart from the cuteness-factor, he opted for “nappies”, because”diapers” could refer to adults too, i. e. lead to misunderstandings in this sentence?
I would class it with words like garbage, as far as use is concerned. The Brits take out the rubbish, we take out the garbage; at the risk of being cute, I wouldn’t trash either; it’s more a matter of dialectic preference, like a Michigander taking his groceries home in a sack, and a New Jerseyan taking them home in a bag. Both words are used in both lects, with different nuances. Then of course, in AmerE there is the joy of redundancy for the sake of rhyme: “Take out the garbage and the trash, or you don’t get no spending cash.”
The word “diaper” has an additional meaning. In design it signifies regular repeating patterns such as are often found in tile work. These are called “diaper patterns”. I prefer to use “nappy” as there is then no confusion with tile work. I am English. It just seems ridiculous to me to call something that derived from a cloth napkin (diminutive, nappy), a diaper.
“… seems ridiculous to me to call something that derived from a cloth napkin…”
Not if (as I believe was the case) napkins commonly had diaper work patterns on them it doesn’t. It’s an easy step to go from, “Please hand me a diaper-work napkin.” To either, “…hand me a diaper.” or “…hand me a napkin.” It’s only surprising (to me) that diaper doesn’t have an equivalent to nappy.
Bricks as well:
http://www.house-design-coffee.com/brick-pattern.html
To avoid confusion I should have written, “…that diaper doesn’t have a short form like nappy.”
This reminded me of the old comedy sketch about getting your nappies or diapers in a twist over an order for double damask dinner napkins: https://www.youtube.com/watch?v=0P8XSUGSR-c