Assuming your are confining your question to health and disease, a diagnostician is one who determines the source and cause of a disease or illness. In principle, this could be anyone. An intelligent layman can observe someone laying on the ground with bone poking through the skin of the leg and the normally straight limb is bent at an unusual angle - and will probably correctly diagnose a fracture. However, your question presumably is concerned with which professionals may be considered diagnosticians in a social or public sense. Historically only medical practitioners have claimed that title - with good cause. Over the last 50 years or so, the expansion of knowledge in the health sciences has been so great that no one person could hope to be competent across the full breadth of the science. Medical specialization has addressed much of the problem, but so has the upskilling and development of paramedical professions and their development of specialists. Nurses, physiotherapists, podiatrists, dentists and others are also diagnosticians now - within their areas of training (undergraduate or post graduate).  In my view, any health professional with an appropriate evidence based training in the science of diagnosis should be able to claim the title - within the area of their expertise.

I agree with Laslett, considering the real meaning of diagnosis and the training and practice of some other health professionals. i look forward to see more contributions