[Ruling on one who performed wuḍūʾ or ghusl and missed the time]

Answer (and Allah grants success): The one who wakes up must perform wuḍūʾ, or ghusl if ghusl is due, and pray.
If it is said: “That will make him miss the time
,”We reply: The time is not considered missed for such a person. In the well-known report from the Prophet (May Allah bless him and his family and grant them peace): “Whoever sleeps through a prayer or forgets it, let him pray it when he remembers it.” It is not permissible to leave wuḍūʾ or ghusl except where using water is difficult due to illness or its absence.
The proof is the verse of wuḍūʾ: “O you who have believed, when you rise to [perform] prayer, wash your faces …” [Al-Māʾidah:6], and His saying: “So fear Allah as much as you are able.” [At-Taghābun:16]
As for the adhān and iqāmah, they are omitted if one fears missing the time by performing them. This may be inferred from the fact that the time has become exclusively for performing the prayer itself, and because neither adhān nor iqāmah is a condition for the prayer’s validity.
Source : Min Thimār al-ʿIlm wa al-Ḥikmah vol.1