Is it permissible to cut ties with the wife's sister who lives with a Christian without marriage and does not wish to leave him, for fear for the sons and daughters?

The default rule is that severing ties of kinship ( قطع الرحم ) is forbidden. However, it is permissible if there is a legitimate religious benefit ( مصلحة شرعية ), and it may even become obligatory if maintaining those ties leads to corruption ( فساد ) or discord ( فتنة ). It is not permissible for a Muslim woman to marry a Christian, by consensus ( بالإجماع ). You must advise her and instill in her the fear of Allah.