GW may be a British company, but Amazon is an American company, the author is American, and they are asserting a trademark violation of something a US author wrote, and tried to sell through a US company to a US audience.
So, unless the British Intellectual Property Office has jurisdiction in the United States that I am unaware of, GW would need to go through the United States Patent and Trademark Office to register a trademark stateside.
Nobody has registered a trademark anywhere, as I think this thread makes clear.
And Amazon is an international company with corporate headquarters in multiple countries (and, more importantly, tax jurisidictions). If you get to say your HQ is in Dublin for tax reasons, you don't get to say you're an "American company". And saying "oooh, this one in America/Timbuktu/The Cayman Islands is our real HQ, not those other ones" is utterly meaningless in the modern age for anything other than propaganda.
Internationals don't work like that.
[Also, while I am certainly not a lawyer and know that we have lawyers here (S'mon and dannyalcatraz come to mind, and S'mon is specifically an expert on British IP law IIRC so he may be the best ENWorlder to weigh in on this issue), I was under the impression that common law trademark is pretty weak IP protection.
Claiming a common law trademark on a phrase that can clearly be seen as in use in an entire literary genre dating to over 50 years before your company ever used it, and has been in significant use in literature and other creative works like video games since you used it sounds like a recipe for failure. Note that GW only tried to use it against a small-time independent author, not against any of the larger companies to use the term, but thanks to the Internet, this little shakedown attempt became big news.
Yup, it's weak. Hell, it's pretty much invalid, let alone weak. You're preaching to the converted. Nobody's defended GW here.