You can simply declare myString as a class member and get the value by a getter method and pass it to your function:
QString Dialog::getString()
{
return myString;
}
void MainWindow::on_dnldButton_clicked()
{
//Dialog is showed via "Modal approach".
Dialog newDnld;
newDnld.setModal(true);
newDnld.exec();
dnldFile(newDnld.getString());
}