The parts of the body that must be washed in wudoo’ are mentioned in the verse in which Allah, may He be exalted, says (interpretation of the meaning): “O you who believe! When you intend to offer As-Salat (the prayer), wash your faces and your hands (forearms) up to the elbows, rub (by passing wet hands over) your heads, and (wash) your feet up to ankles” [al-Maa’idah 5:6].
Allah, may He be exalted, has made it obligatory to wash the arms up to the elbows after washing the face, and this cannot be done except by washing the arms from the fingers up to the elbows. The one who limits it to washing them from the wrists to the elbows has not performed this obligation.
With regard to washing their hands at the beginning of wudoo’, this is Sunnah, and cannot suffice for the obligatory action, according to the majority of scholars, contrary to the Hanafi view.
The majority of scholars are of the opinion that it is obligatory to wash the parts of the body in wudoo’ in the correct order as mentioned in the verse: washing the face, then washing the arms, then wiping over the head, then washing the feet.
Based on that, it is not valid to regard washing the hands at the beginning of wudoo’ as being sufficient and not washing them again with the rest of the of the arm, because that leads to failure to follow the proper sequence, washing of the face in the middle of washing the arms. What is required is to wash the entire arm, after washing the face.
To sum up: if a person does wudoo’ and washes his hands, then rinses his mouth and nose, and washes his face, then he washes his arms from the wrists to the elbows, his wudoo’ is not valid according to most of the scholars.
Shaykh Ibn Jibreen (may Allah preserve him) was asked: What is the ruling on one who washes his arms from the wrist to the elbow, without washing his hands, thinking that his washing of his hands at the beginning of wudoo’ is sufficient? Does he have to repeat wudoo’?
He replied: It is not permissible when doing wudoo’ to wash the forearm only, without the hand. Rather when he has finished washing his face, he should start to wash the arms, and he should wash each of them from the fingertips up to the elbow, even if he washed his hands before his face, because washing them the first time is Sunnah, and washing them after the face is obligatory. If a person only washes his arms from the wrist to the elbow, then he has not completed wudoo’ as required, and he has to repeat his wudoo’ after completing it, or he has to wash what he omitted if only a short time has passed (since he completed it), so he should wash the hands and what comes after that.
End quote from al-Lu’lu’ al-Makeen min Fataawa Shaykh Ibn Jibreen, p. 77
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: Here we should point out something that many people neglect, as they wash the arm from the wrist to the elbow, thinking that they had finished washing the hands before washing the face, but this is not correct. It is essential to wash the arm from the fingertips to the elbows.
End quote from al-Liqa’ ash-Shahri, 3/330
And Allah knows best.