Is the prayer valid for someone who unintentionally missed the Fajr prayer, whether they woke up late and were unable to pray before the time expired, or they woke up early and unintentionally fell asleep until the time expired?

The prayer is valid in both cases if its conditions of validity are met. It is considered performance (ada'an) in the first scenario if what constitutes a prayer is caught before sunrise. As for the second scenario, it is a make-up (qada'), and there is no sin in either case if there was no negligence, due to the Prophet's (peace be upon him) saying: "There is no negligence in sleep; negligence is only on him who has not prayed a prayer until the time for the next prayer comes. So whoever does that, let him pray it when he remembers it," and "Whoever forgets a prayer or sleeps through it, its expiation is to pray it when he remembers it."