Pollard [1], Nicholson [2], and Schonhage-Strassen [3] seem to have come up with it independently around the same time, using different approaches. Strassen is said to have discovered the Pollard approach in 1968 but there is no (written) record of it.
It should also be noted that, while it was not exactly the birth of the FFT, Cooley-Tukey's 1965 paper [4] on it was what kickstarted research on FFT and its applications. This was just a few years after that.