Any-to-any singing voice conversion is confronted with a significant challenge of ``timbre leakage'' issue caused by inadequate disentanglement between the content and the speaker timbre. To address this issue, this study introduces a novel neural concatenative singing voice conversion (NeuCoSVC) framework. The NeuCoSVC framework comprises a self-supervised learning (SSL) representation extractor, a neural harmonic signal generator, and a waveform synthesizer. Specifically, the SSL extractor condenses the audio into a sequence of fixed-dimensional SSL features. The harmonic signal generator produces both raw and filtered harmonic signals as the pitch information by leveraging a linear time-varying (LTV) filter. Finally, the audio generator reconstructs the audio waveform based on the SSL features, as well as the harmonic signals and the loudness information. During inference, the system performs voice conversion by substituting source SSL features with their nearest counterparts from a matching pool, which comprises SSL representations extracted from the target audio, while the raw harmonic signals and the loudness are extracted from the source audio and are kept unchanged. Since the utilized SSL features in the conversion stage are directly from the target audio, the proposed framework has great potential to address the ``timbre leakage'' issue caused by previous disentanglement-based approaches. Experimental results confirm that the proposed system delivers much better performance than the speaker embedding approach (disentanglement-based) in the context of one-shot SVC across intra-language, cross-language, and cross-domain evaluations.