I see someone who is described as a podiatrist, although she doesn't have a degree (I know this as she often talks about her student daughter being the first in the family to go to university), and she is very good. The first appointment with consultation is £45, and subsequent ones £35 for 20 minutes in a clinic. There is no massage or anything - she cuts my nails, files down the hoof-like ones and removes any hard skin. I assume if I had other problems she would deal with them too.
I also have a Greek toe, and the nail seems as thick as it's long. According to the podiatrist, they thicken as they rub on shoes (even well-fitting ones) as it's Nature's way of protecting the toe from harm. It happens very gradually, so it's more noticeable when we are older.