Orthogonal frequency-division multiplexing (OFDM) has been selected as the basis for the fifth-generation new radio (5G NR) waveform developments. However, effective signal processing tools are needed for enhancing the OFDM spectrum in various advanced transmission scenarios. In earlier work, we have shown that fast-convolution (FC) processing is a very flexible and efficient tool for filtered-OFDM signal generation and receiver-side subband filtering, e.g., for the mixed-numerology scenarios of the 5G NR. FC filtering approximates linear convolution through effective fast Fourier transform (FFT)-based circular convolutions using partly overlapping processing blocks. However, with the continuous overlap-and-save and overlap-and-add processing models with fixed block-size and fixed overlap, the FC-processing blocks cannot be aligned with all OFDM symbols of a transmission frame. Furthermore, 5G NR numerology does not allow to use transform lengths shorter than 128 because this would lead to non-integer cyclic prefix (CP) lengths. In this article, we present new FC-processing schemes which solve or avoid the mentioned limitations. These schemes are based on dynamically adjusting the overlap periods and extrapolating the CP samples, which make it possible to align the FC blocks with each OFDM symbol, even in case of variable CP lengths. This reduces complexity and latency, e.g., in mini-slot transmissions and, as an example, allows to use 16-point transforms in case of a 12-subcarrier-wide subband allocation, greatly reducing the implementation complexity. On the receiver side, the proposed scheme makes it possible to effectively combine cascaded inverse and forward FFT units in FC-filtered OFDM processing. Transform decomposition is used to simplify these computations, leading to significantly reduced implementation complexity in various transmission scenarios. A very extensive set of numerical results is also provided, in terms of the radio-link performance and associated processing complexity.