  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 $f\gtrsim0.03$ 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 ($\propto \nu^{-0.4}$)
than predicted by simple jet models ($\propto \nu^{-1}$), and estimate a jet
opening angle of $\phi\sim0.4-1.8$ 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.
