What is the ruling on and how should one deal with a niece who is married to a Christian man in Britain, and could speaking with her be a cause for her husband's conversion to Islam?

It is not permissible for a Muslim woman to marry a non-Muslim, based on the Almighty's saying: (And do not marry [your women] to idolaters until they believe), and His saying: (And never will Allah grant to the disbelievers a way [to prevail] over the believers). The niece must be advised by explaining Allah's ruling, which necessitates her separation from the non-Muslim man, and that her relationship with him is pure fornication. She can marry him if he embraces Islam, as Umm Sulaym (may Allah be pleased with her) did. The man should be invited to Islam, for "it is better for you that Allah guides one man through you than for you to possess red camels." As for dealing with the niece, the focus must be on her abandoning this grave sin, whether through advice and continued communication, or by boycotting and shunning, depending on what the advisor deems more beneficial.