We present new low-frequency observations of the nearby radio galaxy Fornax A at 154 MHz with the Murchison Widefield Array, microwave flux-density measurements obtained from WMAP and Planck data, and gamma-ray flux densities obtained from Fermi data. We also compile a comprehensive list of previously published images and flux-density measurements at radio, microwave and X-ray energies. A detailed analysis of the spectrum of Fornax A between 154 MHz and 1510 MHz reveals that both radio lobes have a similar spatially-averaged spectral index, and that there exists a steep-spectrum bridge of diffuse emission between the lobes. Taking the spectral index of both lobes to be the same, we model the spectral energy distribution of Fornax A across an energy range spanning eighteen orders of magnitude, to investigate the origin of the X-ray and gamma-ray emission. A standard leptonic model for the production of both the X-rays and gamma-rays by inverse-Compton scattering does not fit the multi-wavelength observations. Our results best support a scenario where the X-rays are produced by inverse-Compton scattering and the gamma-rays are produced primarily by hadronic processes confined to the filamentary structures of the Fornax A lobes.