How should one act upon learning of an illicit relationship between a Muslim woman and a Christian man, given the impossibility of intervening to stop it or discussing it with the Muslim woman concerned?

It is your duty to advise this sister. If she does not accept your advice, you may inform her guardian or the competent authorities. Changing the evil here is done by the tongue, as long as you are unable to change it by hand. If you are unable to do that, then you must denounce it in your heart, which is the weakest of faith.