Is it permissible for us to demand our deceased mother's right from our maternal uncle, even if that leads to severing ties of kinship or resorting to the courts?

You have the right to claim your mother's inheritance, even if it means resorting to the courts, unless your mother had pardoned her brother and absolved him of her right. If she did not absolve him but simply refrained from claiming it, then you have the right to claim your share.