"do Teslas apply OTA updates while the car is underway"
Even your cellphone doesn't update when it's 'under way', what makes you wonder a moving automobile will? If you hazard a guess as to how OTA's are applied to automobiles, I would guess it'll be in Parked, plugged in, and prompted to update by pressing an "Accept & Update" button.
There are realtime kernels that can update the apps, the OS, and install a new kernel without dropping a beat. VisualWorks Smalltalk had a system for loading code into a shadow of the running class hierarchy, then atomically committing the code changes. If you defined some functions for the transformation of existing objects, you could start a transaction in one version, then finish it in another.
We keep forgetting that lots of "advanced" features were actually invented decades ago.
Even your cellphone doesn't update when it's 'under way', what makes you wonder a moving automobile will? If you hazard a guess as to how OTA's are applied to automobiles, I would guess it'll be in Parked, plugged in, and prompted to update by pressing an "Accept & Update" button.