web: how long should I wait before I animate the navigation bar in / out when the user moves?

If it reacts to a user action more than 300 ms later, it will not be perceived as a reaction to the action in the first place.

In other words, there is not a minimum, but a maximum of how long you can wait.

My recommendation would be to wait for nothing: what is the value of keeping the menu bar while the user moves? If you make it clear that the bar is disappearing due to the downward movement (by reacting <300 ms), users may infer that it will reappear when scrolling up.