Hi Tom,
Just curious - if you log out of everything (sharepoint, 365) in Chrome, then clear all cookies in the browser (not just the site) for 'all time', then close and re-open, does it work?
For critical staff I've sent a link to Firefox for Desktop that they can install as a normal user. Also that they need to pin it to their taskbar if they want to find it again!
Thanks,
Paul.