After looking on the web, it appears that there is no single British rule that all British publications follow. If you look at what The Guardian (certainly British) actually does, it seems to change full stops into commas and leave them inside the quotation marks:
When Mike said 'Be careful what you wish for,' Janet listened.
The full stop turns into a comma (because it no longer ends a sentence), but it remains inside the quotation marks, indicating that this corresponds to a pause in the original quotation. This is what I would do, but I am American, and thus shouldn't be trusted on this.
However, some sites advocate that you use:
When Mike said 'Be careful what you wish for', Janet listened.
since there wasn't a comma in the original quotation.
The other answer links to the Oxford University Style Guide. This says
include punctuation which belongs to the quote inside the quotation marks, and a closing full stop/question mark/exclamation mark if the quote is a complete sentence.
If you take this literally as written, you would have to say:
When Mike said 'Be careful what you wish for.' Janet listened.
since 'Be careful what you wish for.' is a complete sentence. This looks wrong to me, but certainly much less wrong than:
*When Mike said 'Be careful what you wish for.', Janet listened.
I would hope nobody uses this last style.
Having in recent years taken an undergraduate degree in history, at a British university, I have had a level of instruction on punctuation, footnoting etc. Below is how I would punctuate your examples.
But let me say from the outset that I am assuming that whether the 'sign' contains a full stop is totally irrelevant to the reader. If it were a matter of great relevance, perhaps a forensic clue to a major crime, I would add footnotes to explain. I certainly wouldn't rely on the reader picking up the correct inference from the subtlety of the punctuation!
The sign said 'Handle with care'.
The sign said 'Handle with care'.
Pete said, 'The sign said "Handle with care" '.
Pete said, 'The sign said "Handle with care"'. [This most definitely has to have the full stop last. The only instance in which I would not put the stop last would be if the whole thing was in quotes.]
"The sign said 'Handle with care'", Mike said. [I would put the comma after the second close quote though I wouldn't be dogmatic about it.]
Best Answer
Yes. But I don't understand why you removed the commas in the examples with dashes.
Dashes can replace semicolons, colons, and ellipses, albeit are considered informal.
Note: they may also be used as parenthesis:--
My homework--all Math-, English-, and Science-related--is due tomorrow. *em-dashes have been used.
My homework (all Math-, English-, and Science-related) is due tomorrow.
My homework, all Math-, English-, and Science-related, is due tomorrow.
All are acceptable methods.