What is the ruling on a woman uncovering her hair and face before a non-Muslim male doctor for the purpose of treatment and out of necessity, especially given the absence of a female specialist in her town?

There is no harm for a woman to go to a male doctor when a female specialist and competent doctor is not available. The conditions for this are: only the affected area should be exposed, and the rest should be covered; a mahram (definition), husband, or a trustworthy woman must be present to prevent seclusion; and the examination must be limited to what is necessary for diagnosis and treatment, without excess.