We demonstrate a wide-band all-optical method of nanoscale magnetic resonance (MR) spectroscopy under ambient conditions. Our method relies on cross-relaxation between a probe spin, the electronic spin of a nitrogen-vacancy centre in diamond, and target spins as the two systems are tuned into resonance. By optically monitoring the spin relaxation time ($T_1$) of the probe spin while varying the amplitude of an applied static magnetic field, a frequency spectrum of the target spin resonances, a $T_1$-MR spectrum, is obtained. As a proof of concept, we measure $T_1$-MR spectra of a small ensemble of $^{14}$N impurities surrounding the probe spin within the diamond, with each impurity comprising an electron spin 1/2 and a nuclear spin 1. The intrinsically large bandwidth of the technique and probe properties allows us to detect both electron spin transitions -- in the GHz range -- and nuclear spin transitions -- in the MHz range -- of the $^{14}$N spin targets. The measured frequencies are found to be in excellent agreement with theoretical expectations, and allow us to infer the hyperfine, quadrupole and gyromagnetic constants of the target spins. Analysis of the strength of the resonances obtained in the $T_1$-MR spectrum reveals that the electron spin transitions are probed via dipole interactions, while the nuclear spin resonances are dramatically enhanced by hyperfine coupling and an electron-mediated process. Finally, we investigate theoretically the possibility of performing $T_1$-MR spectroscopy on nuclear spins without hyperfine interaction and predict single-proton sensitivity using current technology. This work establishes $T_1$-MR as a simple yet powerful technique for nanoscale MR spectroscopy, with broadband capability and a projected sensitivity down to the single nuclear spin level.
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