What is the ruling on someone who intended to perform 'Umrah but was prevented from doing so due to an excuse, and then appointed someone else to perform it on their behalf?

It is not permissible for someone who has entered into the state of ihram for the rituals of Hajj or Umrah to exit from it or cut it short before completing them. If your father is able to return to Mecca to complete the Umrah, he must do so. However, if there is a valid excuse preventing him from completing it, then he is considered muhsar (hindered). In this case, he should sacrifice an animal (hady) at the place where he was hindered or in Mecca, then shave his head and exit the state of ihram.

Since your father did not complete the Umrah and returned to Medina due to his illness and old age, he must send someone to sacrifice an animal for him in Mecca and distribute it to the poor of the Haram. Then, he should shave or shorten his hair, and he remains in the state of ihram until this is done. However, if he stipulated at the time of entering ihram by saying: "O Allah, my place of exiting ihram is where You have prevented me," then he exits ihram without a sacrifice or shaving.

The Umrah performed by your brother on behalf of your father is valid if your father is unable to perform Umrah due to old age or an illness from which there is no hope of recovery.

As for you, it was obligatory for you to complete your first Umrah upon your return to Mecca, because you remained in the state of ihram. Your second ihram is considered void, and your circumambulation (tawaf), striving (sa'i), and shaving are considered a completion of the first Umrah. If you committed any prohibitions after your return from Mecca and before completing the Umrah, you are required to offer a fidyah (expiation) for each prohibition, unless you were ignorant, in which case you are excused.