Is it obligatory for the wife to intervene to mend the relationship between her husband and his mother, especially since the husband has not spoken to his mother for three months due to her bias towards his unmarried brother and his refusal to organize his personal belongings, which cause clutter in the house, even though the wife fears God's wrath upon her husband because of this estrangement?

What you mentioned about the mother-in-law concerning her son's injustice and deprivation of his rights is considered an impermissible wrongdoing. And what the questioner mentioned about the husband, regarding his blaming, shouting at, and abandoning his mother, is a grave mistake and a major sin. For the rights of parents are among the most emphasized rights after the right of Allah Almighty, and the mother, in particular, holds a great status. The Prophet, peace and blessings be upon him, made filial disobedience one of the gravest major sins after associating partners with Allah. Abandoning and boycotting one's mother is among the highest degrees of disobedience, which is severely and strictly forbidden. It is incumbent upon the husband to hasten to repent from his disobedience to his mother and seek her forgiveness.