Hmm. I'd be surprised if it's reminding you to get the phone, as that's not something that it ever does. I walk away form my phone all the time with no beeping. I'm guessing it's something else.
The fact that it's happening with two different apps (Workouts & Nike Run Club) makes me think that it's not something to do with the app itself.
Aside from the possibility that you have some other app installed that does this (alert you when you walk away from your phone), the only thing I can think of is that somehow your workouts aren't registering as started & you're getting the Start Workout Reminder. For testing purposes, you might want to go into the workout link in the watch app on your phone, and uncheck Start Workout Reminder & End Workout Reminder (and also Running Auto Pause since you're in there) to see if that solves your problem.