This is especially annoying if you're listening to streaming radio (like NPR) and going for a walk. It will just drop. This has been the case since the very first version of the cellular watch, I suspect as a battery-saving feature.
The only way to really get around this is to force cellular mode before you leave. This is what I do, and it's embarrassingly un-Apple in experience:
(1) Put my iPhone in Airplane mode. I think I could accomplish the same thing just turning off BT, but I want to be really sure the phone and watch are cut off from each other.
(2) Turn WiFi off on my Apple Watch. Wait a few minutes(!) before I go for a walk -- make coffee, leash up my dog, etc
(3) Walk outside, and issue my Siri command to trigger music/radio
This usually works fine. But its annoying. If you're not especially dedicated to going phone-free (it drains the battery quickly) I really wouldn't bother.
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.