I have created a Qt application with Mingw. The application runs properly on the development computer after copying the required DLL's to the directory. However, when deploying on external computers, the application does not run at all.
Update: In addition, the application uses Sqlite3 via QtSql.
It's quite likely you're still missing some dll.
Another thought is some missing configuration file or registry settings.
Double check against the Qt docs for deploying Windows Qt applications.