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.
The ruling on this matter is that a Muslim should perform ablution in accordance to what is stated in the Quran and the way the Prophet Muhammad performed ablution.
Allah Says (what means): {O you who have believed, when you rise to [perform] prayer, wash your faces and your forearms to the elbows and wipe over your heads and wash your feet to the ankles.} [Quran 5:6]
Al-Bukhari and Muslim reported the way the Prophet used to perform ablution, that he used to begin with washing his two hands three times, then he rinses his mouth three times, rinses his nose by inhaling water three times, washes his face three times, washes his forearms up to the elbows three times, wipes his head by passing his hands (after wetting them) over it by starting from the front part backwards then returning to the front, wipes his ears (inside and outside), and finally he washes his feet up to the ankles.
There is no blame on washing each organ one time, or two times since such options were all practiced by the Prophet as authentically reported.
The aforementioned order in which ablution is to be done is a matter of Sunnah (recommended and rewarded) but not a matter of obligation. The preponderant opinion on this issue is the opinion adopted by the Maaliki School of jurisprudence that it is an obligation for those who remember it (the order) or are able to do it conveniently, but it is not an obligation on those who forget it or are not able to do it.
Imam Maalik was once asked, as in the Mudawwanah, about the one who performs ablution without the right order, so he washes his feet before his hands, then he washes his face, and then he performs prayer. He replied: “His prayer is valid.”
As a conclusion, your prayer is valid, Allah willing, but you have to abide by the order stated in the above verse and the Prophet's practice.
For more benefit on the continuation when performing ablution, please refer to Fatwa 139853.
For more benefit on the detailed description of ablution, please refer to Fataawa 82845 and 83164.
For more benefit on obligatory acts that must be washed in ablution, and which are detailed in sequence order, please refer to Fatwa 326854.
Allah knows best.