Time-Scale Modification (TSM) of speech aims to alter the playback rate of audio without changing its pitch. While classical methods like Waveform Similarity-based Overlap-Add (WSOLA) provide strong baselines, they often introduce artifacts under non-stationary or extreme stretching conditions. We propose STSM-FILM - a fully neural architecture that incorporates Feature-Wise Linear Modulation (FiLM) to condition the model on a continuous speed factor. By supervising the network using WSOLA-generated outputs, STSM-FILM learns to mimic alignment and synthesis behaviors while benefiting from representations learned through deep learning. We explore four encoder-decoder variants: STFT-HiFiGAN, WavLM-HiFiGAN, Whisper-HiFiGAN, and EnCodec, and demonstrate that STSM-FILM is capable of producing perceptually consistent outputs across a wide range of time-scaling factors. Overall, our results demonstrate the potential of FiLM-based conditioning to improve the generalization and flexibility of neural TSM models.