Pokémon go won't start on mobile data

Check to see if 'Restrict app background data' is turned on.

To do that for an individual app: Go to Settings -> Apps -> [The App] -> Data Usage

There is a little slider that says 'Restrict app background data'. Make sure it is turned off for Pokemon Go, Google Play Store and Google Play Services.

Make sure they are all turned off. In my personal experience, this solved the problem of not being able to log in without WiFi.

Since Pokemon GO is working with WiFi it must work with mobile data too. This can not be the applications fault, the problem must lie in connectivity of 3G.

May I ask, perhaps you have limited connection? Meaning you have used all the data you had for that month or other time period and now you're stuck with very slow down/up speeds.

If that's not the case then you can try to use the same SIM card in another mobile phone to figure out if your 3G causes the problem.

You can also do a quick speed test to figure out the speeds for your connection.