This paper presents a low-latency real-time (LLRT) non-parallel voice conversion (VC) framework based on cyclic variational autoencoder (CycleVAE) and multiband WaveRNN with data-driven linear prediction (MWDLP). CycleVAE is a robust non-parallel multispeaker spectral model, which utilizes a speaker-independent latent space and a speaker-dependent code to generate reconstructed/converted spectral features given the spectral features of an input speaker. On the other hand, MWDLP is an efficient and a high-quality neural vocoder that can handle multispeaker data and generate speech waveform for LLRT applications with CPU. To accommodate LLRT constraint with CPU, we propose a novel CycleVAE framework that utilizes mel-spectrogram as spectral features and is built with a sparse network architecture. Further, to improve the modeling performance, we also propose a novel fine-tuning procedure that refines the frame-rate CycleVAE network by utilizing the waveform loss from the MWDLP network. The experimental results demonstrate that the proposed framework achieves high-performance VC, while allowing for LLRT usage with a single-core of $2.1$--$2.7$ GHz CPU on a real-time factor of $0.87$--$0.95$, including input/output, feature extraction, on a frame shift of $10$ ms, a window length of $27.5$ ms, and $2$ lookup frames.