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.
All the scholars of Islam are agreed upon the lawfulness of wiping on the cast instead of washing the organ that is injured. This wiping is done in ablution, Ghusl (ritual bath) and Tayammum. The evidence for this is the Hadeeth reported from ‘Ali that he said: “My forearm got broken during the battle of Uhud and the banner fell off my hand. The Prophet told me: “Put the banner in your left hand. You are the holder of my banner in this life and the Hereafter.” I said: “O Prophet of Allah! What do I do with the cast.” He said: “Wipe on it.” [Ibn Maajah]
It is a condition that the broken or wounded organ cannot bear to be washed, or that washing could hurt it; in which case, it should only be wiped. Washing the healthy organ will not cause any harm to the unhealthy person otherwise he should do Tayammum (dry ablution) to avoid worsening the situation of the unhealthy organ. The ruling of wiping applies as long as there is a cast on the organ no matter how long that takes.
For more benefit, please refer to Fatwa 419824.
Allah knows best.