Is the questioner obligated to intervene to reconcile her mother and maternal uncle, and what is your advice to her after her mother refused to initiate the greeting, given her fear of renewed disagreement and backbiting if she intervenes?

The mother and the maternal uncle are relatives whose ties must be maintained, and severing them is forbidden. You have done well by striving to reconcile them, so continue doing so, seeking Allah’s help and supplicating frequently. You can also seek assistance from some respected figures to explain to them the importance of kinship and the obligation to maintain its ties, and that Satan strives to create discord among loved ones. Furthermore, the Sharia (definition) has permitted lying for the sake of reconciliation due to the overriding benefit it entails.