Out of mild curiosity--since I am keen on inscribed books in a vague, general way-- I've occasionally looked for Dunhill books that have been signed. And my impression is that they are uncommon, regardless of which member of the Dunhill family might interest you.
As for this copy and its signature, I'd lean towards the "Nots."
While Alfred the younger (1896-1971) was both the author of this particular book, and indeed still alive, judging from a 1939 letter that I have a scan of, his signature was very different from his father's. See below.
On the other hand, what appears here seems to be loosely modeled on that of the elder Dunhill's very stylish one. This was actually used as a 1922 trademark by the company, and also appears--black on black-- on the slipcover of the 1992 coffee table book by Michael Balfour,
Alfred Dunhill: One Hudred Years and More. See below.
I could of course be entirely wrong. But, it's at least a start on answering your riddle.