We present multi-frequency monitoring observations of the black hole X-ray binary V404 Cygni throughout its June 2015 outburst. Our data set includes radio and mm/sub-mm photometry, taken with the Karl G. Jansky Very Large Array, Arc-Minute MicroKelvin Imager Large Array, Sub-millimeter Array, James Clerk Maxwell Telescope, and the Northern Extended Millimetre Array, combined with publicly available infrared, optical, UV, and X-ray measurements. With these data, we report detailed diagnostics of the spectral and variability properties of the jet emission observed during different stages of this outburst. These diagnostics show that emission from discrete jet ejecta dominated the jet emission during the brightest stages of the outburst. We find that the ejecta became fainter, slower, less frequent, and less energetic, before the emission transitioned (over 1-2 days) to being dominated by a compact jet, as the outburst decayed toward quiescence. While the broad-band spectrum of this compact jet showed very little evolution throughout the outburst decay (with the optically thick to thin synchrotron jet spectral break residing in the near-infrared/optical bands; ∼ 2 − 5 × 10 14 Hz), the emission still remained intermittently variable at mm/sub-mm frequencies. Additionally, we present a comparison between the radio jet emission throughout the 2015 and previous 1989 outbursts, confirming that the radio emission in the 2015 outburst decayed significantly faster than in 1989. Lastly, we detail our sub-mm observations taken during the December 2015 mini-outburst of V404 Cygni, which demonstrate that, similar to the main outburst, the source was likely launching jet ejecta during this short period of renewed activity.