Is it permissible for my wife to give birth in a hospital where all the doctors are non-Muslim men, or should we go to a more distant and difficult hospital where there are non-Muslim female doctors?

A female doctor must oversee the childbirth due to the need to expose the 'awrah and for physical contact. If she is not available, or if reaching her involves hardship and difficulty, then there is no harm in a male doctor, even if he is a non-Muslim, overseeing the childbirth, due to necessity.