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, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
Washing the hands at the beginning of the ablution is an act of the Sunnah (it is not an essential part of ablution); however, this does not mean that the person does not have to wash them again when washing his arms, which is one of the obligations of ablution. If he did not wash his hands when he is washing his arms and settled for washing them at the beginning of ablution only, then his ablution is incomplete. Also, if you did not wash your face, then you have left out one of the agreed-upon obligations of the ablution. The prayer is invalid when the ablution is missing one of its obligations. This is the scholarly view adopted at Islamweb.
However, some scholars held that washing the hands at the beginning of ablution is enough. The Hanafi scholars held that it is sufficient to wash the hands at the beginning of ablution and that there is no need to wash them again with the arms, regardless of whether the person washed the hands with the intention of carrying out an obligation of ablution or not. We hope that there is a way out for you in this view as far as your past ablutions is concerned, and you have to make sure to wash the hands with the arms in the future.
Even based on the view that the ablution is invalid when performed as such, according to a group of scholars, you are not obliged to make up for your past prayers. This was the view chosen by Shaykhul-Islam Ibn Taymiyyah who held that when a person neglects one of the obligations or pillars out of ignorance, he is not required to repeat the past invalid acts of worship.
Shaykhul-Islam Ibn Taymiyyah said:
"As for the person who is ignorant of the obligation in this regard, he should observe it starting from the current prayer and onwards, and he is not required to repeat the past invalid prayers. It was narrated in an authentic report that when the Prophet, sallallaahu ‘alayhi wa sallam, saw a man pray too quickly, he said to him, 'Go back and pray, for you have not prayed…' [Al-Bukhaari and Muslim] The Prophet, sallallaahu ‘alayhi wa sallam, instructed him to repeat that prayer that was currently due, but he did not instruct him to repeat all his previous prayers because he had been ignorant of the correct manner of praying, although apparently he had been praying in this manner in the past. The man said, 'By Him Who has sent you with the truth, I do not know a better way of praying.' Due to his ignorance, the Prophet, sallallaahu ‘alayhi wa sallam, excused him for those past occasions but he commanded him to repeat the prayer that was currently due. Moreover, he did not command ‘Umar to repeat the prayers when he became Junub (ritually impure after sexual activity), nor ‘Ammaar when he rolled in the dust, nor Abu Tharr for what he left of prayers while he was Junub. He did not instruct the woman experiencing vaginal bleeding outside the menstrual period to repeat all her past prayers although she stated that she experienced long bleeding that prevented her from performing the fasting and prayers. He did not order those who used to eat in the nights of Ramadan until the white thread of dawn would become distinct from the black thread of night to repeat their fast. At first, the obligatory prayer was only a two-Rak‘ah (unit of prayer) prayer, and it became a four-Rak‘ah prayer in residence after the migration to Madeenah. Many Muslims in Makkah, Abyssinia, and the desert did not become aware of that change except after a period of time. He, sallallaahu ‘alayhi wa sallam, did not command them to repeat all their prayers that they had performed during this period. In addition, he did not instruct the Muslims who used to pray in the direction of Al-Aqsa Mosque, before knowing of the abrogation in this regard and of the change of the Qiblah, to repeat all their past prayers that they performed before knowing of the change of the Qiblah."
As for your Tawaaf (circumambulation), ‘Umrah, and Hajj, they are valid as well according to a group of scholars, including Shaykhul-Islam Ibn Taymiyyah, as they held that ritual purity is obligatory for Tawaaf but that it is not one of the conditions for its validity. You should make sure in the future, as we mentioned above, to wash the whole hands with the arms.
Allah knows best.