No, you can use AirPods on a non-cellular watch just fine, it has bluetooth. Cellular only affects your ability to stream content (eg. Music or podcasts), but you are also able to download/cache those while on WiFi on the non-cellular watch.
>As in, if I leave my phone at home, will I be able to stream music and calls from my watch to my AirPods while I go for a run?
For my response, I assumed two things:
(1)"Streaming music" meant listening to music that is not stored on the device.
(2) "Calls from my watch to AirPods" meant that during a run, OP was not in constant range of a remembered, open WiFi network.