What's the mechanical 'whoosh, whoosh' sound in this video?
It's the sound of the trim wheel rotating on the center console of the Boeing 737.

The trim wheel controls elevator trim. It can either be moved automatically by the autopilot, or manually by the pilot. The pilot can push the trim button on the yoke with their thumb (left for the captain, right for the copilot).



This is a photo of a B737 classic series, but this feature is unchanged on the NG series.

In this specific instance, it sounds like there were two small manual push on the trim button.