Do I have to be a nurse to then become a wound care technician?

Wound professionals. It is not clear what you mean by a "wound technician". There are multiple opportunities with wound care under many different professions.
Yes and no. Many employees at our wound care center are non-nursing trained. Nursing, although is an excellent long term career option with many levels of educational achievement. Nursing aid, lpn, rn and doctorates of nursing. Some wound care certifications may require an rn designation.
Not necessarily. There is a difference in being a nurse and a wound care technician. The level of training is different and the responsibilities are different depending where you work.