Will the spouses be sinful if they allow their mother-in-law, who prostrates to other than God, to do so in their home, and is it obligatory for them to prevent her from doing so for fear of God's wrath and its impact on the children?

Filial piety is an individual obligation (fard ayn). One must be dutiful and kind to their mother, even if she is not Muslim, unless she commands an act of disobedience or polytheism. This is based on the Almighty's saying: "Allah does not forbid you from those who do not fight you because of religion and do not expel you from your homes - from being righteous toward them and acting justly toward them." You must speak to her gently and kindly, avoid showing annoyance, advise her with kindness, and invite her to Islam. As for the polytheistic acts she performs privately, there is no blame on you for them, and you are not obligated to forcibly prevent her. Rather, you should continue inviting her to Islam with gentleness and kindness, and not despair of her guidance. As for the children, their beliefs must be safeguarded, and faith must be instilled in their hearts. The grandmother's error should be explained to them without leading them to dislike her.