The app doesn't care if it's mobile or WiFi, LTE or HSPA or EDGE.
You need to talk to your mobile provider and ask if they allow any internet traffic, or just web browsing (and what looks like web browsing, e.g. cloud based mail apps like MyMail, BlueMail, Outlook, etc.).
As a sanity check, feel free to try K9 Mail, it's free in Play.
A debug log too would help shed light on this.