I don’t think Fleming believed the Cold War would end by 1961, but he did believe that SMERSH was losing popular appeal as the epitome of evil. This was because of the changing political climate in the late 50’s, as Khrushchev’s "thaw" improved relations between the USSR and the western world. Fleming’s villains were always larger than life and twice as evil, but the public’s image of the Soviet Union was undergoing significant changes.
After Stalin died in 1953, Khrushchev began a program of de-Stalinization that culminated in his speech denouncing Stalin before the Communist Party Congress in 1956. Khrushchev criticized the Great Purge in which millions of citizens were killed, millions more sent to the gulags. There was a power struggle within the Kremlin. Khrushchev kicked out a lot of pro-Stalinists, released political prisoners (the gulag population went down from around 13 million to 5 million during his rule), posthumously "rehabilitated" the reputations of dissenters murdered during Stalin’s regime.
Along with these political changes, Khrushchev also liberalized the Soviet Union. There was a sort of cultural renaissance in this period, with whole new waves of music, literature, cinema emerging; that had previously been repressed. He reached out to the western world, organizing cultural and sports exchanges, unbanning lots of western books, sending Soviet artists abroad and inviting western artists to the USSR. He personally approved the prize when American Van Cliburn won the Tchaikovsky Competition in Moscow in ’56.
He was also a lot friendlier towards the US. He attended peace conferences, elaborated his doctrine of "peaceful coexistence", met with Eisenhower, eventually visited the US in 1959. So with all this happening, the western public was losing its view of the Soviets as the ultimate enemy. I don’t know if they thought the Cold War would soon be over, but SMERSH was definitely becoming a harder sell.
It’s not just the thawing of US-Soviet relations that made him replace SMERSH with SPECTRE. Fleming was British, and he wrote his books during the post-war period when the British Empire was falling apart. His books are full of comments, both from Bond and from foreigners, that Britain was just "giving it all away", failing to defend its power and prestige. In his early books, he maintains the fiction that Britain is still the key player in world affairs, with Bond coming in to rescue the bumbling Americans from their own incompetence, with British ships and resources showing up to fix situations the Americans can’t handle. But this was also getting to be a harder sell, and Fleming’s attitude towards the US changes from a friendlier "we’re co-equals in this fight against communism" to a less friendly attitude verging towards contempt, as the US began to dominate and Britain’s star faded.
In this situation, he needed a new villain. Cold War villains were unfashionable. Portraying the British Secret Service as playing a key role in world affairs was less believable. So he settled on the apolitical criminal organization SPECTRE in Thunderball, with Ernst Stavro Blofeld as the criminal mastermind.
Of course, the thaw in the Cold War was only temporary. Thunderball was published in March 1961. Between April 1961 and November 1962, there was the Bay of Pigs invasion, the Berlin Wall went up, there was the Cuban Missile Crisis. The political atmosphere changed quite drastically, and perhaps Fleming could have resurrected the Soviets as villains again. But it was becoming less credible to have a British spy play the critical role in confounding the Russians, that role now went to the Americans. The success of Ernst Stavro Blofeld in particular as an acceptable villain persuaded Fleming to reuse him in two more books: On Her Majesty’s Secret Service and You Only Live Twice. Fleming died in 1964, so he never got the chance to get back to the Russians even if he had wanted to.
A good source for this stuff is the book The World of James Bond: The Lives and Times of 007 by historian Jeremy Black.