Google Account Log in doesn't work on LTE/Data
I had the exact same problem: Cannot log in to google account when on data (works on Wifi)

After contacting both google and my network provider, it was clear that the problem is on the device. For me there was a built-in security app from Lenovo, blocking 'google services'. 

Check all your firewalls, antivirus programs and the settings>data that nothing as googleservices or googleframework is blocked. Uncheck 'restrict background data' in case it is enabled for all apps or google-specific. Also allow syncing data for apps while on data ofcourse.