What it really means is naff… (you do know what a dag is? It isn’t positive :D) Usage might be a bit light=hearted, but it’s not a positive descriptor.
UK: High street.
US: Shopping district.
US by a mile. It means what it says. Game over.
A variation of the game:
UK: Top-shelf.
US: Top-shelf.
For the US, it means top of the line, best, grade-A. In the UK it means where the porno mags are displayed in a store (and therefore material of that nature in general).
US wins again because, but just by a bit, due to the meaning being a bit more obvious.
This is another one of those “be careful in using it in front of people on the other side of the pond” things. E.g., “That celeb’s memoir is really top-shelf.”
What is jaggery, you ask? It’s solidified molasses. Not raw sugar, but wholefood sugar.
UK (or at least England): “That’s the thing, innit?”
US: “Exactly”.
I give a slight edge to the US because it’s shorter and clearer. The few times I’ve heard someone in England say “That’s the thing, innit?” to express an agreement so strong with what I just said that nothing further needs to be added, I instinctively wait for them to keep talking because in America when we say “that’s the thing, isn’t it?” we are also expressing agreement but then usually continue to explain why we agree.
On the other hand the UK expression is more euphonious.
I’m in Aus, and building owners have given up on that stuff. Level 1 is the sub-floor services, Ground floor on the downhill side is maybe level 3, and on the uphill side maybe level 5, Plumbers go to the level as directed, and occupants just have to deal with it.
Aus: High street.
Well in Melbourne, High street, the main street, is going to run along the ridge. That’s why they put the shops there, out of the mud.
I’m giving the edge to the UK, simply because I like the sound of “innit”.
Brit
Rocket
US
Arugula
Result: Tie
Reason: Who cares?
Brit
Skirting Board
US
Baseboard
Result Brit
Reason: It just sounds better, even if baseboard is more descriptive
Brit
Austin Powers
US
Wayne Campbell
Result: Brit
Reason: Throw in Dr. Evil and it’s a no-brainer
Brit
Disorientated
US
Disoriented
Result: US
Reason: Disorientated sounds like some ridiculous, American bastardization and not the other way around.
Mmmm… not convinced. 
Yep, know what they are, and have even used “rattle yer dags” on occasion… but my experience (YMMV of course) is that “he’s a bit of a dag” is rather more blokey, rural, bit of a hard-case, and so on… than “naff”.
But that’s a different phrase to “daggy” and dag can be used in varying contexts and tones, but something that is daggy is “uncool”, unfashionable, daggy, naff
Just to be awkward, it’s a subway in Glasgow (although you do hear some people calling it “tube”, but they are silly). “Underground” or “subway” is OK, and escapes my wrath.
Fair enuff. 
US: speeding*
Aus: hooning
Australia clearly wins.
(I may or may not be muttering “hooooooon” under my breath and giggling now.)
- yes I’m aware this isn’t the most direct analog to “hooning” but I am brainfried at the moment and others can probably suggest better analogs than I.
UK: Wendy house
US: playhouse
Result: US. “Wendy house” gets points for the Peter Pan reference but loses points for being twee*, while “playhouse” is just a more descriptive and comprehensible term.
UK: twee
US: cutesy
Result: UK. “Twee” is a great word. Unfortunately, its use is effectively restricted to Brits because it’s kind of twee for non-Brits to use it.
We call them cubby houses. Not very comprehensible if you don’t know what it is.
You neglect “fanny”, for which the American terms are “trim” or “camel toe”. Neither side should get a win.
I’m not sure that “fanny” and “camel toe” are the same thing, are they? Surely “fanny” relates to “vulva”, while “camel toe” is more about the visual appearance of the front when very tight clothing is worn.
I confess I had not previously heard of “trim” in this context, but I’ll be careful not to go to a hairdresser shop and ask for a trim.
It makes about as much sense as does “Wendy house”. I think the Americans win this one. 
Only if you use it as a playhouse. My cubby house is home to my chooks.
Well, flats are usually all on one level with no internal stairs.
As for this “story” business, in the UK, it means, well, a story. A tale. In terms of referring to whichever level of a building, it is “storey”.
Tangentially, I seem to recall reading references to a “cold water flat” and something like a “railway carriage flat”. I might be misremebering the second one. But what are those about? 
The usage may not be that widespread, but “a trim” is perfectly safe (most of the time), but you should avoid saying “some trim”.