Firstly, what you've posted isn't an example of British punctuation: there's no such thing. The style of punctuation depends on the style guide you're using, and this is often dictated by the area you're writing for (e.g. scientific vs. creative). A friend of mine studied Geography and had to use, Harvard referencing and punctuation inside the quotes. When he did his masters (in Creative Writing) he had to learn to use single quotes, MHLA referencing and punctuation outside the quotes.
Punctuation should only go outside the quotes if it's a quote - not dialogue. If you're writing an essay in British English and quoting from a book, that's when the punctuation goes outside the quotation marks. However, many people agree that this looks ugly and a few people from my MA course (in Creative Writing) only used this because they had to.
Single quotes vs. double quotes varies from person to person. At school I was taught to use double quotes. When I did my MA, we were encouraged to use single (but could use double if we wished). The main reason we were told to use single was because double quotes inside single looks better than single inside double. A lot of the time, punctuation is as much about how it looks on the page as what its function is.
Punctuation between British and American English really isn't that different. The main issues are spelling and different words for things (for instance, biscuits, cookies and scones mean different things depending on if you speak to someone English or American). I'd be much more concerned about that than punctuation because it's far more difficult to spot and easier to get confused.