Is the wife obligated to emigrate by herself from Western countries to Algeria, given her husband's opposition and his arguments about the unavailability of housing, and in light of her fear of losing her children if she were to do so?

A wife's obedience to her husband in what is right is obligatory, and this includes that he should not command her to disobey Allah. It is not permissible for a woman to obey her husband in permanently residing in disbelieving lands without a compelling necessity, as this is a great sin. If the husband does not object to returning but wishes to prepare suitable conditions, then her obedience to him in waiting is not considered disobedience, provided that the waiting period does not become so prolonged that the children are exposed to trials (fitna). If the children's exposure to trials is imminent, then it is obligatory to leave with them immediately. If the husband insists on staying after trials appear, then Allah and His Messenger are more deserving of obedience, and the wife must arrange for the emigration of herself and her children.