Nanophotonic spiking neural networks (SNNs) based on neuron-like excitable subwavelength (submicrometre) devices are of key importance for realizing brain-inspired, power-efficient artificial intelligence (AI) systems with high degree of parallelism and energy efficiency. Despite significant advances in neuromorphic photonics, compact and efficient nanophotonic elements for spiking signal emission and detection, as required for spike-based computation, remain largely unexplored. In this invited perspective, we outline the main challenges, early achievements, and opportunities toward a key-enabling photonic neuro-architecture using III-V/Si integrated spiking nodes based on nanoscale resonant tunnelling diodes (nanoRTDs) with folded negative differential resistance. We utilize nanoRTDs as nonlinear artificial neurons capable of spiking at high-speeds. We discuss the prospects for monolithic integration of nanoRTDs with nanoscale light-emitting diodes and nanolaser diodes, and nanophotodetectors to realize neuron emitter and receiver spiking nodes, respectively. Such layout would have a small footprint, fast operation, and low power consumption, all key requirements for efficient nano-optoelectronic spiking operation. We discuss how silicon photonics interconnects, integrated photorefractive interconnects, and 3D waveguide polymeric interconnections can be used for interconnecting the emitter-receiver spiking photonic neural nodes. Finally, using numerical simulations of artificial neuron models, we present spike-based spatio-temporal learning methods for applications in relevant AI-based functional tasks, such as image pattern recognition, edge detection, and SNNs for inference and learning. Future developments in neuromorphic spiking photonic nanocircuits, as outlined here, will significantly boost the processing and transmission capabilities of next-generation nanophotonic spike-based neuromorphic architectures for energy-efficient AI applications. This perspective paper is a result of the European Union funded research project ChipAI in the frame of the Horizon 2020 Future and Emerging Technologies Open programme.