Deep neural networks (DNNs) enhance the accuracy and efficiency of reconstructing key parameters from time-resolved photon arrival signals recorded by single-photon detectors. However, the performance of conventional backpropagation-based DNNs is highly dependent on various parameters of the optical setup and biological samples under examination, necessitating frequent network retraining, either through transfer learning or from scratch. Newly collected data must also be stored and transferred to a high-performance GPU server for retraining, introducing latency and storage overhead. To address these challenges, we propose an online training algorithm based on a One-Sided Jacobi rotation-based Online Sequential Extreme Learning Machine (OSOS-ELM). We fully exploit parallelism in executing OSOS-ELM on a heterogeneous FPGA with integrated ARM cores. Extensive evaluations of OSOS-ELM and OSELM demonstrate that both achieve comparable accuracy across different network dimensions (i.e., input, hidden, and output layers), while OSOS-ELM proves to be more hardware-efficient. By leveraging the parallelism of OSOS-ELM, we implement a holistic computing prototype on a Xilinx ZCU104 FPGA, which integrates a multi-core CPU and programmable logic fabric. We validate our approach through three case studies involving single-photon signal analysis: sensing through fog using commercial single-photon LiDAR, fluorescence lifetime estimation in FLIM, and blood flow index reconstruction in DCS, all utilizing one-dimensional data encoded from photonic signals. From a hardware perspective, we optimize the OSOS-ELM workload by employing multi-tasked processing on ARM CPU cores and pipelined execution on the FPGA's logic fabric. We also implement our OSOS-ELM on the NVIDIA Jetson Xavier NX GPU to comprehensively investigate its computing performance on another type of heterogeneous computing platform.