Aside from not seeing any motion at a tangent, how fast was the ship rotating?
Is the wrong question.
The question is "How far are you from the axis of rotation?" and also "Exactly how much sideways force were all those people experiencing in those scenes?" To the latter question, probably less than you'd think; they're falling way too slowly for that, and Chekov's "gotcha" moment on the catwalk sort of hints they were being pulled sideways at something like one third of a gee or less.
But let's be generous and call it a full G. If they were only a hundred meters from the ship's axis of rotation, that would be consistent with a turn rate of about 18 degrees per second; that is, a rate at which it would take the ship a full ten seconds to tumble 180 degrees. That, IMO, is quite a bit slower than the ship appears to be tumbling; OTOH, the catwalk is in engineering, probably much closer to the ship's center of gravity, so the rotational velocity at that moment could easily be twice that.