This paper introduces a novel "all-spike" low-power solution for remote wireless inference that is based on neuromorphic sensing, Impulse Radio (IR), and Spiking Neural Networks (SNNs). In the proposed system, event-driven neuromorphic sensors produce asynchronous time-encoded data streams that are encoded by an SNN, whose output spiking signals are pulse modulated via IR and transmitted over general frequence-selective channels; while the receiver's inputs are obtained via hard detection of the received signals and fed to an SNN for classification. We introduce an end-to-end training procedure that treats the cascade of encoder, channel, and decoder as a probabilistic SNN-based autoencoder that implements Joint Source-Channel Coding (JSCC). The proposed system, termed NeuroJSCC, is compared to conventional synchronous frame-based and uncoded transmissions in terms of latency and accuracy. The experiments confirm that the proposed end-to-end neuromorphic edge architecture provides a promising framework for efficient and low-latency remote sensing, communication, and inference.