Carrie-Anne by The Hollies. Mentions “prefects”.
Peaches The Stranglers
*A Girl Like You * The Troggs
Come Up and See Me (Make Me Smile) Steve Harley and Cockney Rebel
When I Was Young The Animals
Appetite Prefab Sprout
And anything by the Arctic Monkeys, the Housemartins or [cough] Chas and Dave.
Greenday tries to sound British, even though there a wholly American band.
Exactly the first thing I thougt of! Slight hijack: how many people actually say"supernover" when singing along? I do, every time.
Lazy Sunday by The Small Faces
National Express by the Divine Comedy. (National Express is the British equivalent of the Greyhound buses).
Common People by Pulp - which additionally IMO is the best pop song ever written, no question.
I think “It Would be so Nice” is a very British sounding song.
As is “Time”.
Must be the quiet desparation?
Syd’s Floyd was very English, I’m thinking of songs like Scarecrow and The Gnome.
My offering - Bike.
or “She’s Bought a Hat Like Princess Marina’s”
You’re definitely right, and it’s because he’s half-speaking and half-singing in this song. More leaning towards speaking. The British accent comes out way more in speech than in singing. The song’s not really about Cowboys and Indians (just the opening line,) it’s about a guy who watches too much TV and therefore his actual life seems unglamorous in comparison. The structure of the song takes the form of four vignettes from a guy’s life, told in the first person: the first, he describes watching a Cowboys and Indians show; in the second, he’s watching The Sweeney; in the third, he goes to a bar, gets drunk, makes an idiot of himself; the fourth, he picks up a woman at a disco, sleeps with her, leaves, and then basically reflects on how the experience was disappointing (“it’s not like that on the TV when it’s cool for cats.”)
One of the best Squeeze songs, and one of the more unique pop songs out there in general. I grew up listening to Singles: 45s and Under and am a big fan of the group. I might be the youngest Squeeze fan on earth.
London Calling — The Clash
The Kinks again and Lola.
Yep, I didn’t reckon for the Guthrie and Wilco songs that he’s done more recently. More of the early, I’m-forming-Red-Wedge-stuff.
Hell, we could put up “Tracy Jacks,” “Bank Holiday,” or “This Is A Low” as well. Civil servants wigging out and streaking, the unheard-of-in-America holiday celebration, and the shipping forecast for the English Channel. But I would also argue that Modern Life Is Rubbish (despite the inclusion of “Miss America”) and The Great Escape are pretty English albums as well.
Let’s add Morrissey to the list: “Irish Blood, English Heart,” “Come Back To Camden,” “Dagenham Dave,” “Hairdresser On Fire,” etc.
I would also suggest that the 2-Tone bands like The Specials and Bad Manners had a very British sound, even though they were merging West Indian music with punk.
Came in to mention exactly that!
The Proclaimers are very very Scottish (which is good:) ) and they belong in here because the title says “British” not “English.” I second the mention of the Pistols (“Anarchy in the U.K.” “God Save the Queen”) and the Clash.
The exact 2 songs I was thinking of (along with Billy Bragg, just because of his accent).
While the aforementioned songs by The Kinks were the first things to come to my mind as “very British sounding rock songs” (“Victoria”, “Waterloo Sunset”), there’s also Paul McCartney’s song Uncle Albert/Admiral Halsey.
It sounds like Sir Paul was smoking some serious pipe when he wrote that song, but damn it, it was British weed!
The Kinks’ ‘Dedicated Follower of Fashion’ (1965) mentions Regent St, Leicester Sq, and, obliquely, Carnaby Street, upon all of which I will probably trod (as well as Waterloo Bridge) when I’m back in the ‘Big Black Smoke’ (1966) in about two weeks. 
‘Holiday in Waikiki’ is also very English in its own way. So many of their [Ray’s] songs are essentially English – ‘House in the Country’, ‘Rainy Day in June’, ‘Most Exclusive Residence for Sale’, ‘Dead End Street’, ‘Mr Pleasant’, ‘David Watts,’ ‘Tin Soldier Man,’ ‘Autumn Almanac’ and so on up through Ray Davies’ current stuff, like ‘Next Door Neighbour’ and ‘Stand Up Comic’.
Ray’s stuff tends to be far more English (a rather wistful and nostalgic English) than Dave’s, but it’s Dave who penned ‘Dear Margaret’.
Allow me to mention Def Leppard . Maybe not so much for the sound, but the fact that they go out of their way to accentuate their nationality–the Union Jack is displayed prominently on their clothes, their concert set, their albums.
As far as sound, I’d go with “Run Runaway” by Slade , “Does Your Chewing Gum Lose its Flavor” by Lonnie Donegan, most songs by The Cure (especially “Just Like Heaven”),and almost anything by Depeche Mode .
And just about everything by The Who . “A Quick One” is particularly British.