I’m not a royalist that much is true but I don’t hate the old dear, not at all.
Brenda BTW is the name used for the Queen in Private Eye Magazine
I assume you would count your accent as context? If an obvious Brit were to speak of “the Queen”, I would of course assume it was Elizabeth being referenced. As others have said, it’s hard to imagine a realistic situation where someone would refer to Elizabeth II with so little context that you wouldn’t infer the reference, so for all practical purposes, the answer to your questions is yes, people will make that assumption.
That said, if a guy I’ve never seen before walks up to me on the street and says, with the accent of Joe Midwestern American, something like “I like the Queen”, then my first reaction would be to wonder what he was talking about. If I had to guess without being able to ask any questions, I would probably say Elizabeth just for lack of obvious other options. I would think it more likely, though, that he was talking about some locally-known person called “the queen” of whom I was unaware, similar to the way a San Franciscan might reference the Emperor, or a resident of a city that just held an annual ceremony might reference the ceremonial “queen”.
I’d assume he was dining alone. 
In quotes like the thread title, that’s what I assumed too, as in: “How did that Helen Mirren movie do at the American box office?”
I personally would assume any reference to “The Queen” (I’d lowercase “the” unless it was at the beginning of the sentence, although I know that’s not how Buck House does it) would be to Her Britannic Majesty (as she is designated on British passports, to differentiate her from all the other queens out there), Elizabeth II. She is commonly referred to by the American media as the “Queen of England,” although of course she’s actually queen of the United Kingdom of Great Britain and Northern Ireland (of which England is but one important part).
No, Kevin Rudd, a Labour politico, is the new Australian PM. Took office Dec. 3, has already reversed Howard’s policy on global warming and the Kyoto Protocol, but says he still wants close ties to the U.S. Interestingly, at least according to Wiki, he didn’t swear allegiance to the Queen of Australia (also Elizabeth II), but swore to faithfully serve “the people of the Commonwealth of Australia.” So I guess he’s not much of a monarchist.
I can’t imagine an American thinking of anyone but Elizabeth II when hearing/reading the phrase “The Queen,” entirely without any context.
And if hearing “The Queen is dead,” for a second or two I’d assume Elizabeth II, but then I’d ask “You do mean Elizabeth of England, right?”
Now if you said “The Queen is pregnant,” my first instinct would be to think Liz has had some miraculous new medical treatment, and intends to finally acquire a child she can be proud of . . . but after a few seconds I’d have to ask “Which Queen?”
I can’t believe I never heard anything about this?
Do you read the papers? I know that last November’s election result was covered in the Canadian press because I got emails from friends commenting on it.
The only time I wouldn’t think that referred to QE II of the UK might be if it was said in a gas station. Then I might think of Queen Beatrice of the Netherlands, owner of most of Royal Dutch Shell Oil company.
Ah, but the OP defines his own context, doesn’t he? Being English, he automatically defines himself as talking about QE2, just as one of our Dutch Dopers would similarly define himself as talking about Queen Beatrix.
Yes, almost everyone would think first of Elizabeth II. If you said, "No, not that Queen, we’d wander through Aretha Franklin, the Queen of Soul; KoKo Taylor, the Queen of The Blues; Leona Helmsley, the Queen of Mean; and a few other lesser royalty. Very few of us know of any other actual monarchic Queens.
If I heard someone say something like “The Queen is visiting with the President this week”, I would immediately assume that it was Elizabeth II, Queen of Far Too Many Places to List, rather than any of the other current monarchs or consorts. I would also assume that the President referenced is George W. Bush (or whoever is the President of the United States at the time), if I heard the statement made in the US. But if I were anywhere in the world where they don’t have their own queen, Liz it is.
Now, if it were a context (or lack thereof) that did not imply a major geopolitical figure, then I might assume otherwise. If, for instance, I were on the phone with my mom, and she said “Hang on, the Queen wants something”, I would probably assume that she was referring to Lil, one of the two cats who lives with her (or whoever happens to be the dominant housecat at the time).
He asked about the U.S. though.
**You know what the funniest thing about other parts of Europe are?
Jules Winnfield:
What?
Vincent Vega:
It’s the little differences. I mean they got the same sh** over there that they got here, but it’s just, just there it’s a little different.
Jules Winnfield:
Example.
Vincent Vega:
The have kings and queens. Most of them aren’t even the Queen of England
Jules Winnfield:
They don’t call them the queen of England?
Vincent Vega:
No, man, they got the metric system, they don’t know what the fu** a Queen of England is.
Jules Winnfield:
What do they call it?
Vincent Vega:
They call it a Royal family with Cheese.
Jules Winnfield:
Royal family with Cheese.
Vincent Vega:
That’s right.
Jules Winnfield:
What do they call a president?
Vincent Vega:
presidents is a presidente, but they call it Président de la République .
Jules Winnfield:
Le Big Mac. What do they call a Prime Minster?
Vincent Vega:
I don’t know. I didn’t go into Parliament.**
No, no, no. Morrissey surely?
Shagnasty, that was just great!