What you need to know is here, it was Fleming himself who suggested the change whilst working for producer Kevin McClory on the script for what would become Thunderball (book and film) prior to making the first film Dr No:
Reflections on the Origins of SPECTRE
In a memo dated 15 June 1959 Fleming wrote that it would be unwise to directly identify Russia as the enemy: ‘Since the film will take about two years to produce, and peace might conceivably break out in the meantime, this should be avoided’. He suggested instead that the villainous plot be the work of an international terrorist group known by the acronym SPECTRE.
Fleming claims that he closed down SMERSH in his novels and invented SPECTRE to reflect what he saw as a partial thawing of the Cold War. (Bond himself admits in Thunderball that ‘with the Cold War easing off, it was not like the old days’.) The reality was that, as Fleming knew, Britain’s ability to act as an effective force in the international struggle against ‘Redland’ was, by this time, stretching the bounds of credulity. Fleming’s Playboy interview also suggested a literary logic behind the change, proposing that SPECTRE represented a ‘much more elastic fictional device’. Put simply, the new organisation appeared to offer more wide-ranging potential for an engaging thriller narrative, not least because the master-criminal or super-villain now truly represented a universal, international threat to humanity. He is no longer simply an organ or agent of state. Blofeld’s organisation, we are told, is a ‘private enterprise for private profit’ – in the parlance of modern intelligence agencies SPECTRE is a potent, hostile ‘non-state actor’.