Is it permissible for the grandmother to remarry, given that her grandfather did not officially divorce her until after he discovered that the divorce papers had not been finalized?

If the grandfather divorces his wife, the grandmother, whether by an explicit declaration of divorce or by an implicit word with the intention of divorce, then the divorce takes effect. Her marriage to another man after the expiration of her waiting period (iddah) is valid. The lack of official registration of the divorce does not affect its validity or occurrence.