With as little detail as posted - no idea.
In general, the app doesn't care what the network connection is, as long as it works.
When the phone is connected to mobile - does the "last sync time" below each account change to red? If so, that means the app is trying to check mail but is running into network errors.
What happens if you try to "refresh" manually when on the mobile connection? Does it work? If not, any error messages?
Finally where are your accounts - Gmail, local ISP, Yahoo, Hotmail, are they all the same (in this sense) or different?