We present simultaneous multi-band radio and X-ray observations of the black hole X-ray binary Cygnus X-1, taken with the Karl G. Jansky Very Large Array and the Nuclear Spectroscopic Telescope Array. With these data, we detect clear flux variability consistent with emission from a variable compact jet. To probe how the variability signal propagates down the jet flow, we perform detailed timing analyses of our data. We find that the radio jet emission shows no significant power at Fourier frequencies $fgtrsim0.03$ Hz (below $sim30$ sec timescales), and that the higher frequency radio bands (9/11 GHz) are strongly correlated over a range of timescales, displaying a roughly constant time lag with Fourier frequency of a few tens of seconds. However, in the lower frequency radio bands (2.5/3.5 GHz) we find a significant loss of coherence over the same range of timescales. Further, we detect a correlation between the X-ray/radio emission, measuring time lags between the X-ray/radio bands on the order of tens of minutes. We use these lags to solve for the compact jet speed, finding that the Cyg X-1 jet is more relativistic than usually assumed for compact jets, where $beta=0.92^{+0.03}_{-0.06}$, ($Gamma=2.59^{+0.79}_{-0.61}$). Lastly, we constrain how the jet size scale changes with frequency, finding a shallower relation ($propto u^{-0.4}$) than predicted by simple jet models ($propto u^{-1}$), and estimate a jet opening angle of $phisim0.4-1.8$ degrees. With this study, we have developed observational techniques designed to overcome the challenges of radio timing analyses and created the tools needed to connect rapid radio jet variability properties to internal jet physics.
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