I actually received a slip in the mail today and went and picked up my parcel. So, either the strike isn't affecting Speed Services, or my parcel managed to be sent before the strike affected CT (not sure if it is localised to CT and JHB). I had assumed the Speed Services tracking was just dodgy, but it was just Kalahari being Kalahari and getting the tracking number wrong. When I put in the tracking number that was actually on the parcel, it worked fine.
So, if, for whatever reason, you have stuff shipped via the postal option (it is only available on some things), you may have issues tracking your parcel.