If you are looking along your centerline/0/180 degrees/backwards, then you can be moving forward at speed. Slow or fast shouldn't matter. At any other direction your speed will drag the periscope line sideways and mess up your target speed measurement.
Think of your periscope line as a wall projected through the water. No matter which angle the target passes through that, it still will take the same time. As long as the wall does not move sideways.
I don't know specific periscope models, but at least some used to have a mechanism in there that had a vertical line indicator that turned slaved to the gyro-compass. Within a limited bearing range. So it kept pointing to a specific true bearing. Germans called this 'Feste Linie im Raum.', in English this would translate to something like Fixed Line through space. A simple form of inertial reference, at least bearing-wise. Years ago I learned this from a website of a person working at a German naval museum. But I think I've also read references to this on the
www.tvre.org/en site. This would have made it easy for them to measure this with the boat turning around on the waves.
I knew it:
http://www.tvre.org/en/stabilized-azimuth-line