Am I sinful, and what is the expiation after I swore by Allah that my wife would not give birth in her city, then she gave birth there before her due date?

You bear no sin for your wife giving birth in her country, nor for merely swearing that she would not give birth in her family's city, and no expiation is required of you. This is because her giving birth in that manner was beyond your control, resembling duress. Thus, if someone swears not to do something, or that his wife will not do something, and then the sworn-upon act occurs under duress, the one who swore does not break his oath.