Processing occurs at both ends of the chain - FMC and Tower,
Mathematical algorithm called a Kalman filter is used and takes several seconds to produce the 'live' readout. A simplified definition of the algorithm would be the combination of information in the presence of uncertainty. It's particularly effective in resolving vector calculations but can take 7 or 8 seconds to produce the result.
You've probably been asked at some point for a 'spot wind' by ATC and the same request from your end can often produce an straight readout without any of the data massaging and inherent delays caused by the kalman filtering.
Rob