Quite often – particularly in the media – I see British examples of things being referred to "A Very British… " whatever it is
e.g. a documentary about the British building their first nuclear bomb was called "A Very British Bomb" – but, OK it was British in that it was made in Britain, but I couldn't see what was "very" about it… (It didn't spray tea and crumpets everywhere or anything)
I don't see it applied to other nationalities (e.g. you wouldn't really see "A Very French… " or "A Very Irish… ")
A quick google search for "A Very British" shows up loads of examples, but noticeably a book from 1982 called "A Very British Coup" – are they all spawning from that or is it just another example?
Edit
To clarify, I'm not just talking about the words "very British" but more the commonality of the use of "A Very British" (I'm not sure of how I'd explain – perhaps that needs another question!) almost as a title or a self-contained grouping of words.
For example: If I asked you where does the common usage of the words "Big Brother" come from, you might say George Orwell's 1984 (if not the TV show! But even there it originates with 1984) – it has a specific use outside just a brother who's big. You probably wouldn't tell me there's nothing special about "big brother" just because a google ngram shows that "little brother" is used more than "big brother"
Best Answer
I think that they are, intentionally or unintentionally, referencing that book, which was also made into a 1988
BBCChannel 4 TV series.https://en.wikipedia.org/wiki/A_Very_British_Coup
When the coup is described as "very British", I believe that they mean "non violent and polite", compared to, for example, military coups in South America which are often very violent and disruptive. In contrast, the "coup" of the title merely involves political machinations involving the civil service and newspapers, via a publishing magnate.
So, to describe something else as "A very British X" carries the implication that it is non-violent and relatively "gentlemanly" compared to what one might normally expect.
In the case of "A very British bomb", I agree that apart from being made in Britain, I don't think there's anything stereotypically British about it, and I don't think it really deserves the implications I refer to above: I don't see how a British nuclear bomb could be more "polite" than an American one for example. I think it's just a bit of lazy titling.
Alternately, the phrase may actually just be taken to mean "Something happening in Britain which you wouldn't normally associate with Britain."
As an aside, the plot of the novel actually seems very similar to what is happening with Jeremy Corbyn, the head of the Labour Party, at the moment, and if Rupert Murdoch started colluding with the civil service to bring him down then it would actually mirror the plot perfectly. We'll have to wait and see...