Prayer is the only pillar in Islam that cannot be waived under any circumstances, unless a person is mentally handicapped or unconscious. It has to be performed whether a person is traveling, sick, poor, rich, married, pregnant, old and so on, unlike other pillars (fasting, pilgrimage, and zakah) that can be waived under certain conditions. There is no provision in Islam to miss any Prayers. Even if you are sick and lying in bed in a hospital without being able to move, you have to pray with your eyes. This shows how essential Prayer is and how no one is allowed to downplay its importance.
If one misses a Prayer out of forgetfulness of sleeping, he/she has to perform it once he/she remembers or wakes up. Sheikh M. S. Al-Munajjid, a prominent Saudi Muslim lecturer and author, stated the following: “Whoever misses an obligatory Prayer because of sleep or forgetfulness has to make up for it when he/she remembers it, because the Prophet (peace and blessings be upon him) said, “Whoever forgets a prayer or sleeps and misses it, let him pray it when he remembers it, for there is no other expiation for it than that” (Al-Bukhari and Muslim).”
Having stated the above, it becomes clear that if a Muslim misses Fajr Prayer, then he/she must pray it straight away without waiting till Zhuhr Prayer.