Should a Muslim perform a complete ablution (wudu') then perform tayammum for the injured part, or should he only perform tayammum if there is a wound on one of the parts of ablution?

If a wound is on one of the ablutionary organs, it is either exposed or covered with a bandage/dressing:

1. If it is covered with a bandage or dressing: The healthy part is washed, then the hand is moistened with water and wiped over the bandage/dressing. There is no need for tayammum (dry ablution). This has been authentically reported from Ibn Umar, may Allah be pleased with him.

2. If the wound is exposed: It is obligatory to wash it with water if possible. If washing is not possible but wiping is, then wiping is obligatory. If neither wiping nor washing is possible, the wound is left without washing or wiping, and then tayammum is performed after completing the ablution. It is not a condition that the bandage or dressing be applied while in a state of ritual purity (tahara).