How can one pray and perform ablution (wudu') when there is a bandage on the the nose, blood is flowing from it, and it cannot be touched?

It is obligatory for you to wash the limbs of ablution that you are able to wash. As for what you are unable to wash—such as an injured nose—you must wipe over it. If you are unable to wipe over it, then wipe over the bandage placed on it, because it is considered like a splint. If you are unable to do all of that, then perform tayammum for the wound that you are unable to wash, wipe over, or wipe over what is placed on it. It is more cautious for the tayammum to be performed when washing the face.

Shaykh Ibn Uthaymeen said: An exposed wound should be washed with water. If that is not possible, then wipe over it. If that is not possible, then perform tayammum. If it is covered with something that is permissible to cover it with, then only wipe over it. If wiping over it while covered harms it, then resort to tayammum.

Regarding tayammum for a wound, Shaykh Ibn Uthaymeen said: If the tayammum is for minor impurity, then the madhhab stipulates sequence and continuity. However, some scholars said that neither sequence nor continuity is stipulated, and this is the correct view and what people practice today.