Former Infantry Officer here with 2 combat deployments and lots of field training. We were never restricted in bringing phones or other electronic devices with us. Some restrictions in the TOC of course, but out in the field no issue bringing stuff out with you. Obviously people couldn't sit there playing with their phones while doing actual training, but as anyone who has been in the military knows, there is a hell of a lot of down time.
I am currently in the NG and we just started AT. I'm heading out in a couple days to the field and will let everyone know how the watch holds up. I did notice the stainless steel gets scratched fairly easily, so I bought a few skins to put on the watch. I don't think there will be any issues, but we will see.