Saltzman bought the rights to the Bond novels for a period of only 6 months in 1961. Saltzman and Broccoli had to get a movie made fast and approached United Artists. David Pickard at UA fostered the deal and encouraged the selection of Dr No because it could be made cheaply and would serve to establish the style for a series.
Guide to British Cinema by Geoff Mayer has more detail about the making of the movie.
There were a number of legal issues surrounding both the novel and screenplay for Thunderball, it’s worth pointing out. It’s also worth noting that the supervillain Blofeld (introduced, along with his organization SPECTRE, in the Thunderball book and an element of the legal battle) isn’t in the novels Dr. No or From Russia With Love but does appear (sort-of) in the film adaptations, probably because Broccoli recognized the value of an archenemy.