The only real justification for seeing dialog as "American" is that there are many words where US spelling seems more "logical" than British - largely thanks to Webster, though so far as I know he never addressed this particular issue.
Paraphrasing grammarist.com's entry on another such word: all early editions of Noah Webster’s dictionary list catalogue, but by the 1890s catalog was commonplace in AmE texts (Webster often gets credit for changes he played no direct part in).
There aren't many written instances of dialog prior its rapid uptake by software-oriented writers, but here's one from 1910 Transactions and proceedings of the American Philological Association, where one might reasonably suppose the author to be perfectly literate.
I think it's just that on average people involved in technical writing are more inclined to ignore precedent and go for what seems to them the logical spelling. Consequently we see the short form much more often in computer contexts such as dialog box, leading many people to suppose there are in fact two different words involved (or at least, that the word has two different spellings dependent on context).
It seems to me usage is currently in a "transition phase", but most likely the shorter form will continue to encroach further into traditional, non-computer contexts, and will eventually be seen as standard for all contexts (but for most people, particularly Brits, this hasn't happened yet).
I'm British - and definitely getting to be the wrong side of "middle-aged", which may have a bearing. But so far as I'm concerned, slut primarily means the first definition in OED...
- a woman of dirty, slovenly, or untidy habits or appearance
Nor do I disagree with their second definition...
- a woman of a low or loose character; a bold or impudent girl; a hussy, jade
That second definition could include anything from a woman who (provocatively or carelessly) shows more bare flesh than is proper, to one who indulges in casual sex more often than most.
But I personally would never extend it to mean "a women who trades sex for money" except where all the above attributes applied anyway, and were more relevant to my context.
But all such words have not only an inherent range of meanings in and of themselves; different people have widely differing value judgements concerning cleanliness, tidiness, sexuality, etc. Not to mention which there are the figurative and facetious usages. Many a male has been accused of whoring himself to [whatever the writer disapproves of].
Use of sexually-loaded terms is also significantly influenced by "publicly-acknowledged morality", as portrayed in the media, and that can vary widely by country. I've recently watched several American "Coming-of-age / High school/college" movies. It took me a while to realise that in that context, blow-jobs are usually seen as "normal, heavy petting", whereas penetrative sex is usually classed as "whoring" regardless of whether anyone actually paid for it. That's Hollywood for you. British movies don't often feature oral sex - but if they do, it's more like "getting to fourth base" (where third base is intercourse).
EDIT: I prefaced this answer by pointing out that it's only a British perspective (which is what OP asked for anyway). But my gut feeling was that younger Brits don't even use slut - they use slag for "promiscuous female or male", and slob for "untidy/dirty/lazy person" (again, unisex).
Google NGrams appears to support my feeling. Check this chart to see how slut has steadily risen in the AmE corpus in recent decades (switch to the BrE corpus and you'll see a corresponding fall). And check this chart to see how she's a slag has gained currency in the BrE corpus (switch to the AmE corpus to see that Americans simply don't use that word often enough to chart usage at all).
It may therefore be true that on the relatively few occasions when younger Brits do come across slut in their native land, they might tend to ascribe it the modern American sense - simply because that's the one they're most likely to have encountered through international media, movies, etc.
Best Answer
Which British English references? The Oxford English Dictionary comments of under way, 'now frequently as one word'. There is a separate entry for underway, and it contains several British citations from the twentieth century.