Any American who wants to get past what you can learn about British words and phrases in a dictionary will benefit from this book.
Divided by a Common Language helped me overcome long-term misunderstandings about what I had been reading in English books. Some British words have an ordinary meaning in American English that is quite different from their British meaning. For example, the British "marrow" is a "large zucchini." For decades, I have been expecting to find beef marrow on my plate in England because of that misreading. I also thought that the British "paraffin" meant a petroleum-based wax as it does in the U.S., whereas it means "kerosene" in the U.K. In reading about someone going for paraffin in novels, I have been wondering what on earth they were going to make with all that wax. If you read this book, you will probably find your own examples of where you thought you knew what was going on . . . but really didn't. I suggest that you start with the British/American Lexicon to learn the most words with the least effort.
The book also has a useful section on British and American phrases, that should help you avoid inadvertently saying what will be perceived as vulgarities across the pond. For example, refer to "retrieving and returning baseballs" rather than "shagging flies" (make your own guess as to what that means, but it isn't nice).
In the vein of the potential for humorous miscommunications, there are a number of cartoons that show what John Bull and Uncle Sam are thinking about when the same word or phrase is said. "My wife loves pot plants, Sam" conjures up John Bull thinking about potted flowers while Uncle Sam imagines a garden full of marijuana plants.
I found four weaknesses in the book that you should be aware of. First, the food equivalents aren't really very precise in some cases. So you may get some surprises. A scone and a biscuit are described as the same, which most of us would argue they are not. A burrito is described as meat and salad in a tortilla, while most would agree that it is as likely to have beans and cheese with sauce as it is to have meat and salad. Second, the book mostly focuses on the contemporary British language so it is less helpful than it might be if you are reading older British books. Third, the U.K. words more represent the usages of English people than they do on what you will encounter in Scotland or Wales. Fourth, some U.K. terms described here are actually in common usage (at least in the northeast and in California, the areas I know best) in the United States. "Pins and needles" is the way many Americans would describe the feeling of blood circulation returning to an limb that has gone numb, but is described as a British term here.
I don't feel competent to review how helpful this book would be to a British person, so I will skip that perspective.
The book covers how the languages came to diverge, tips for tourists, detailed information about important daily subjects (cars, telephones, plumbing, electricity, food, shopping, schools, sights, and measurements), etiquette, driving terminology, pronunciation, spelling, "what not to say," idioms and expressions, and a U.K.-U.S. and a U.S.-U.K. lexicon. There's also some information about Australian, Canadian, New Zealand and South African variations.
Good luck in explaining yourself to those in the U.K.