What is the ruling on continuing to go to a Christian skin specialist for personal care, while fearing to go to others due to Muslim women's lack of experience in this field?

It is not permissible for a woman to expose her major awrah (private parts) to anyone except her husband, or for medical necessity. If skincare requires the exposure of this awrah, then the woman must attend to herself. When there is a need for treatment, a Muslim woman should be prioritized over others, then a trustworthy non-Muslim woman, over a Muslim man. As for what does not require the exposure of the major awrah, it is permissible to seek the help of a Muslim woman, and it is permissible to seek the help of a trustworthy non-Muslim woman, with preference given to the Muslim woman. The fundamental principle is that a woman should attend to herself as much as possible.