Is it permissible for my mother to demand the value of the gold contract she gifted to my uncle and his wife with the intention of helping with an urgent operation, and then it became clear that it was not spent on that, and should she demand it at today's market price for the same number of grams?

Whoever is gifted something to be used for a specific purpose, it is not permissible for him to use it for anything else; due to the saying of the Prophet, peace be upon him: "Muslims are bound by their conditions." So, if your uncle did not use the money from the necklace for the operation, your mother has the right to demand it back, because it was a conditional gift, and it is permissible to revoke it due to the non-fulfillment of the condition.

Your mother can request a necklace similar to hers, or gold by the gram, and she should not ask for money except upon fulfillment, and it is not permissible to agree on money now; because this falls under deferred currency exchange.