All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger. {C}
We hope you are not sinful for not waking up to pray on time.
The Prophet said: "There is no negligence in sleep; rather, negligence is when one does not pray until the time of the next prayer comes. Whoever does that (wakes up late), then let him pray it when he wakes up. However, the next day, he should be keen on performing it at its fixed time." [Muslim]
Since you took the initiative to perform ablution and tried to pray before the time of the prayer expires, then you have done what you are required to do, and you are not required to do anything else. If you had completed one Rak'ah before the time of the prayer expires, then you have prayed on time; otherwise, its time has already elapsed and it is considered as making up for it.
The Prophet said: "Whoever completes one Rak'ah of Fajr before the sun rises, then he prayed the prayer (i.e. prayed the Fajr prayer on time)." [Ahmad and at-Tirmithi]
Allah knows best.