Is it permissible to dispose of the deceased mother's estate by paying Zakat on her behalf from the money she left, without the sister's knowledge, and will the mother's responsibility be absolved after the maternal uncle waives his legal share?

If your mother gave out the remaining amount of zakat that she had miscalculated, then her obligation has been fulfilled, and there is no sin upon her. This is because Islamic law has lifted the burden of error and forgetfulness from this Ummah, as stated in the Almighty's قول: "And there is no blame upon you for that in which you have erred, but [only for] what your hearts have intended."

As for her will concerning some of her heirs, a will made for an heir is religiously forbidden and not enforceable unless the other heirs agree to it. It is considered a form of inflicting harm through a will, which is a major sin. You did well by correcting this mistake, enabling the heirs to take their share, as the Almighty said: "But if one fears from the testator some error or sin and reconciles between them, then there is no sin upon him." If your maternal uncle waives his share, there is no blame upon either of you, and it is hoped that your mother's obligation will be fulfilled thereby.