The growing number of Internet-of-Things (IoT) based artificial intelligence (AI) applications deployed at resource-constrained network edge call for ultra-reliable and low-latency data processing pipelines from distributed front-end sensors to remote inference units. Meanwhile, brain-inspired neuromorphic computing featuring spiking neural networks (SNNs) have arisen as a new paradigm for energy-efficient AI inference. However, significant energy and time expenses incurred in high-complexity transceivers that combat fading and multi-user interference hinder implementations of multi-user neuromorphic inference for edge intelligence. To address this challenge, we consider in this paper a broadband multi-user remote inference system that integrates event-based sensing and time-hopping (TH) on-off keying (OOK) based ultra-wideband (UWB) communications for reliable neuromorphic inference. Specifically, we propose a novel two-timescale repetition coding that leverages intra-frame pulse sparsity for low-latency repetition. We also develop two neuromorphic inference schemes based on: (i) digital spike encoding that recovers each pixel of the event-frame by threshold-adaptive detection via an SNN based sparsity estimator; and (ii) analog spike encoding that converts noisy correlator outputs at the receiver into analog-valued inputs for end-to-end (E2E) classification. Finally, numerical results validate the effectiveness of the proposed coding schemes, and reveal a signal-to-noise ratio (SNR)-dependent performance crossover between the two inference schemes, indicating that analog spike encoding based schemes are preferable with mild or high SNR while digital spike encoding based schemes remain robust in low SNR regime.