Can homoeopath become a radiologist?

I would answer this with a no, because homeopaths are practitioners that work with the homeopathic remedies in a holistic matter.

Radiology is philosophically the opposite of that and it would require a complete new set of training and education.

Technically, they could go on to become a radiologist if they attended four years of medical school and completed a residency in radiology, but I doubt homeopaths do this very often.