I'm not a hydrologist, so I'm not aware of any metrics/heuristics that should likely govern your methods, but here's a stream-of-consciousness response for something that would be fun to try.

First, I'd interpolate points along the line with equal-length, short spacing according to a minimal granularity where the distance between points arguably equates with a distance long-enough to establish a "winding trait". I'd experiment with something like 4x the bank-full width. Or maybe the width of the 25 or 50 yr floodplain as the spacing distance. In other words, something other than a magic-number guess that could be repeated for other streams---doing it that way would make it repeatable for other streams in other studies.

Next, I'd iterate over the points (or do this nested within the interpolation step, above) and get a value for the change in bearing between that point and the points preceding/following it. Obviously, the higher the delta value, the more winding.

Once you had the delta values established, you could probably "eyeball" that data against your flow path and select a thresholds for value ranges that constitute various intensities of winding.

What a fun research question to explore. I'm jealous. :)