For FFT with floating-point numbers, another paper from Arnold Schönhage in 1982 [1] already gives the bound in Psi(n l) operations, where n is the number of coefficients, and l is the desired precision (typically 53 for double precision). Psi(m) is the time to multiply two integers with m digits, which is known since 2021 to be O(m log m) [2]. So the current bound is O(nl log(nl)).

[1]: https://doi.org/10.1007/3-540-11607-9_1

[2]: https://www.texmacs.org/joris/nlogn/nlogn-abs.html