Extracting neural signals at the single motor neuron level provides an optimal control signal for neuroprosthetic applications. However, current algorithms to decompose motor units from high-density electromyography (HD-EMG) are time-consuming and inconsistent, limiting their application to controlled scenarios in a research setting. We introduce MUelim, an algorithm for efficient motor unit decomposition that uses approximate joint diagonalization with a subtractive approach to rapidly identify and refine candidate sources. The algorithm incorporates an extend-lag procedure to augment data for enhanced source separability prior to diagonalization. By systematically iterating and eliminating redundant or noisy sources, MUelim achieves high decomposition accuracy while significantly reducing computational complexity, making it well-suited for real-time applications. We validate MUelim by demonstrating its ability to extract motor units in both simulated and physiological HD-EMG grid data. Across six healthy participants performing ramp and maximum voluntary contraction paradigms, MUelim achieves up to a 36$\times$ speed increase compared to existing state-of-the-art methods while decomposing a similar number of high signal-to-noise sources. Furthermore, we showcase a real-world application of MUelim in a clinical setting in which an individual with spinal cord injury controlled an EMG-driven neuroprosthetic to perform functional tasks. We demonstrate the ability to decode motor intent in real-time using a spiking neural network trained on the decomposed motor unit spike trains to trigger functional electrical stimulation patterns that evoke hand movements during task practice therapy. We show that motor unit-based decoding enables nuanced motor control, highlighting the potential of MUelim to advance assistive neurotechnology and rehabilitation through precise, intention-driven neuroprosthetic systems.