There is this idiom that says "x is as American as apple pie", which means it is very or typically American. I wonder if there is any similar idiom for something that is typical of England or of English people.
There are literally hundreds you could use ("tea and scones", "a seaside town on a rainy day", "talking about the weather", "a bowler hat and a pinstripe suit") but the one you'd choose is always going to depend what other attributes besides Englishness, you want to connote. For example, you wouldn't want to say that the a car was "as English as a rainy day" unless you wanted to indicate that the car was dull; you'd say "as English as a bowler hat, pinstripe suit and an umbrella" (if the car's styling was ultra-conservative), or "as English as a game of cricket on the village green" if it was classic, a bit sportier, but still fundamentally quite comfortable. If you said "as English as a football hooligan in a St George's t-shirt" you'd be saying something quite different about it.
I have heard “as English as a nursery rhyme” and “as English as a rainy day”. Ngram search throws up a few references. Here are two:
“The windscreen wipers were beating a soothing tempo, as English as a nursery rhyme”
”With a name like “Doretti,” you’d think this car would have some Italian sheetmetal or even an Italian engine underneath, but the Swallow is as English as a rainy day”
The allusion to a nursery rhyme probably goes back to the days when Britain was a world power and the English upper and middle classes reared their very young children in a nursery, sometimes with a nurse. Given the British and upper/middle class dominance of the world in Victorian times, the nursery image may well have seemed to typify England rather than other places.
The comparison with a rainy day is less secure. There are many rainier places than England but, nevertheless, many coming to England from sunnier or drier places regarded raininess as typical of England. (Having lived in west Scotland, it is easy for me to surmise that these travellers never went further north to experience Scottish west coast rain!)
The English themselves may resort to cosy thoughts such as:
“Vicars, pillars of the community, as English as tea and cake and cricket on the village green, but times are changing.
Although tea and cake outside in good weather may be common throughout Britain, the mention of cricket distinguishes England from other parts of the country where it was or is less popular.
The only entry of the form "[as] English as" that appears in Frank Wilstach, Dictionary of Similes, second edition (1924) is this one:
English as Piccadilly. — ANON.
A Google Books search finds a dozen or so matches for this expression, from as early as 1882, but mostly from the first half of the twentieth century. An instance from 1900 amplifies the expression to "as English as Piccadilly or the white cliffs of Dover"; and a few Google Books matches—from as early as 1933—deploy "as English as the white cliffs of Dover" on its own.
Elyse Sommer, Similes Dictionary, second edition (2013) offers a greater array of "as British [or English] as" similes:
As British as roast beef —Anon
As British as tea and scones —Elyse Sommer
The variations to this are virtually limitless; to cite just a few: "English as the changing of the guards at Buckingham Palace," "English as clotted cream," "As English as Piccadilly," "As English as Trafalgar Square."
I'm not sure how far to trust Sommer's authority, however, given that (1) she attributes one of these similes to herself; (2) she refers to "the changing of the guards" rather than to "the changing of the guard" at Buckingham Palace; and (3) in her coverage of "as American as," she cites "As American as cheesecake" but not "as American as apple pie."
If you're open to terms referring to British and not English, specifically, there's as British as the Queen which is relatively common. I found several uses in Google Books, and 26,900 hits on Google. You can also find as English as the Queen (88 results on Google for me), but that is less common.
Of course, the term is also not especially accurate since the British monarch isn't particularly British and has ancestors hailing from various other countries. On the other hand, you could argue that this makes the expression even more accurate since modern Britain is a very diverse nation comprised of people from many different ethnic and national backgrounds.
Be that as it may, the expression exists and is in use, albeit nowhere near as common or idiomatic as the American equivalent you mentioned. I can certainly confirm that I have heard it used in the UK today.
https://en.wikipedia.org/wiki/John_Bull John Bull is a national personification of the United Kingdom in general and England in particular, especially in political cartoons and similar graphic works. He is usually depicted as a stout, middle-aged, country-dwelling, jolly and matter-of-fact man.
And more at https://www.historic-uk.com/CultureUK/John-Bull/
Also, in some parts of Britain straight John Bull.