Is it permissible for the questioner's daughter to marry her brother's son, knowing that the questioner and her brother's son were both breastfed by the questioner's sister (though not at the same time)? And would the brother of her brother's son also become an uncle to the questioner's daughter?

If your nephew, your brother's son, nursed from the same sister from whom you nursed, then he is your foster-brother. It is not permissible for him to marry your daughter because he is her maternal uncle through fosterage, and what is forbidden by lineage is also forbidden by fosterage. As for your younger nephew, if he did not nurse from the aforementioned sister, then it is permissible for him to marry your daughter, unless she is forbidden to him due to another fosterage relation.