Dear questioner,
Thank you for your important question.
May Allah, Most High, reward you for your desire to increase in knowledge.
Washing your hands at the beginning of ablution is sufficient, and your previous ablutions are valid.
Washing the hands until the elbows is an integral of ablution. Washing your hands until the wrists, separately at the beginning of ablution, is a Sunna action that ensures no filth on your hands before beginning ablution.
(Shurunbulali, Maraqi al-Falah)
And Allah alone knows best,
[Ustadh] Omar Popal
Checked and Approved by Shaykh Faraz Rabbani