What's the status of the offspring of a Jewish woman who converted to another religion (knowing the woman herself nonetheless remains Jewish)? Are they still Jewish?
Someone born of a Jewish mother remains Jewish whatever they do.
See Can a Jew lose their Jewish Status? which explains why a Jew who tries to convert to another religion is still a Jew though a sinner. As a result, (see "Am I Jewish?") the child of a woman who converts is still Jewish, though the child of a Jewish man with a nonJewish woman is not Jewish, whether the man converted or not.
A Jew remains a Jew, no matter what, even if he converts to another religion. A Jew is defined by someone who is born of a Jewish mother, or someone who converted to Judaism through a proper Orthodox beit din. Therefore, in your case, even though the mother converted to another religion, her children are still Jewish, because she is Jewish herself (that didn't change when she converted, whether she likes it or not). So, her children "receive all the priveledges thereof" of a full-born Jew.
(She herself, too, has all the rights as a full-born Jew, despite her conversion, if she does teshuvah.)
The children of a Jewish man and a non-Jewish woman, however, are NOT Jewish, even if the man didn't convert out of Judaism. Jews must have Jewish mothers. This is the way halacha works.