One possible solution to detect the Nitinol wire passing through the gap is to use an inductive proximity sensor. These sensors work by generating a magnetic field around a coil, and detecting changes in the field caused by the presence of a conductive material (such as the Nitinol wire).
To detect the Nitinol wire passing through the gap, you can place the sensor close to the gap and align it with the path of the wire. As the wire passes through the gap, it will induce a change in the magnetic field around the sensor, which can be detected and processed by the sensor.
Inductive proximity sensors typically have a fast response time, on the order of a few milliseconds, so they should be able to detect the wire passing through the gap in under 100ms.
You can find a range of inductive proximity sensors online or at your local electronics store. Be sure to choose a sensor with a detection range that is appropriate for the size of the gap and the diameter of the Nitinol wire.