Is it permissible for a wife who has undergone a mastectomy due to a cancerous tumor and fears contamination of the wound by water to perform tayammum by having dust placed on her palms, with which she wipes her face, then her forearms and hands, then prays her obligatory prayer, and then performs a regular ablution for other prayers?

What you did was wrong. Your wife is obligated to perform ghusl on the parts of her body that she is able to, and to wipe over the parts that would be harmed by washing. If she is unable to wipe over it, she should wipe over the splint (or bandage) on it. If she is unable to do that, she should perform tayammum.

If the ailment prevents the use of water on all limbs, she should perform tayammum for all of them. If it prevents some limbs but not others, she should wash what is possible and perform tayammum for the rest.

If the wound is exposed, it is obligatory to wash it. If that is difficult, then wipe over it. If that is difficult, then perform tayammum. If it is covered, then only wipe over it. If wiping over it harms it, she should resort to tayammum.

Your wife must make up all the prayers she performed with an invalid purification.