Unfortunately we don’t have One S here, so I can’t try to reproduce it :(. I’ll file a bug and ask around the company if anyone has it.
With the widget I just meant that you should try it if it works. I see that you’ve tried other apps that work, so that’s sorted.
What is weird is that turning mobile data OFF works, but turning it ON doesn’t. Could you try to set the Internet to OFF and see if it behaves the same? Thanks.
Also, what version of Battery Saver do you have? You can find it on the About screen - in “three dot menu” on home screen.