What are the limits of the relationship with the religious Christian mother of one's wife? Should one be wary of her influence on the children, and how should they be treated?

If you marry a girl, her mother becomes your permanent mahram, and her being Christian does not affect this. You must treat her and her children well, as your good treatment of them could be a reason for them to embrace Islam, while bad treatment could alienate them. Maintaining kinship ties is required even if the mother is a disbeliever, as mentioned in the hadith of Asma bint Abi Bakr. However, if you fear that they might cause fitna for your wife or children in their religion, or lead them into sins, there is no harm in taking the necessary measures to protect the religion of your family and children, even if it involves partially or completely preventing them from interacting. Maintaining kinship ties with a disbelieving relative is not obligatory, but it is recommended if their conversion to Islam is hoped for.