Why can't I connect to Pokemon Go via mobile data?

I had the same problem. The way I solved it was to go into my google account settings and allow for less secure apps to access my google account.

Try creating a new account using Pokemon Trainer Club.

There seems to be a routing problem for the google login servers affecting only a small unknown set of people. Although this means you will have to start anew, the game is more fun if you can just walk freely without having do some weird WLAN hopping.