The claim from this book is being published everywhere, inc in the USA, and being widely commented on across all major social media platforms. And overwhelmingly the vast majority of comments are very negative to Harry and megan naming their daughter lillibet, both in terms of the crassness of doing it in the first place, and supporting the queens reported feelings about it. The fact a small minority don’t believe it, doesn’t change that.
its just more negativity about them, on top of so many other things, the interviews, the book, misleading people, like on offering to pay for security, or working for Dior, or the car chase, too many to recount now.
if the queen did give permission and was happy about this, then harry and Megan will take legal action, they are hugely litigious, as this is majorly reputation damaging, to an already damaged nearly beyond repair reputation. If they don’t, then it would validate its true.
as said before, I suspect they will threaten at most, as for everyone to be publishing they must feel the sources are bulletproof, and the last thing Harry and Megan will want is that proven in court.