Is what the questioner and her mother did — giving one of the paternal uncles his share of the inheritance to the exclusion of the other, based on a prior waiver from the uncles — considered a Sharia error, and does the uncle who waived his right still have a right to the inheritance?

There should be no insistence on people to relinquish their legitimate share of inheritance. Such insistence may fall under begging from people, and it contradicts self-respect and being content with what one has, rather than what others possess.

If the paternal uncle willingly relinquishes his share, not out of shyness or compulsion, then there is no harm in you accepting what he has given up.

However, if it is highly probable that he relinquished it out of shyness, then what is taken by the "sword of shyness" has no good or blessing in it. Its ruling is like that of what is taken by coercion. The jurists have unanimously agreed that if something is taken from someone out of shyness, without their full consent, the taker does not rightfully own it, because it is compulsion by the "sword of shyness." In this case, the paternal uncle must be given his share.