In this work, we introduce an optoelectronic spiking artificial neuron capable of operating at ultrafast rates ($\approx$ 100 ps/optical spike) and with low energy consumption ($<$ pJ/spike). The proposed system combines an excitable resonant tunnelling diode (RTD) element exhibiting negative differential conductance, coupled to a nanoscale light source (forming a master node) or a photodetector (forming a receiver node). We study numerically the spiking dynamical responses and information propagation functionality of an interconnected master-receiver RTD node system. Using the key functionality of pulse thresholding and integration, we utilize a single node to classify sequential pulse patterns and perform convolutional functionality for image feature (edge) recognition. We also demonstrate an optically-interconnected spiking neural network model for processing of spatiotemporal data at over 10 Gbps with high inference accuracy. Finally, we demonstrate an off-chip supervised learning approach utilizing spike-timing dependent plasticity for the RTD-enabled photonic spiking neural network. These results demonstrate the potential and viability of RTD spiking nodes for low footprint, low energy, high-speed optoelectronic realization of neuromorphic hardware.