The construction "nobody who ... will..." is equivalent in meaning to "anybody who ... will not...". The negation semantically applies to the clause as a whole, but it is marked on an indefinite pronoun instead of on the verb. This is correct grammar in standard English (although from a comparative standpoint, it's a fairly unusual way to mark clause negation).
The relative clause "who doesn’t support Trump" has a separate negation marked on the verb. This is not ungrammatical; the "error" of using a "double negative" is about using two negative words with a single negative meaning. (Or more than two negative words, as in "Nobody didn't do nothing" = "Nobody did anything"; which is why negative concord is a better term for this phenomenon than "double negative".) The two negative words in "nobody who doesn’t support Trump will ever trust their judgment or patriotism again" have different meanings, serving to negate the main clause and the embedded clause respectively. Compare sentences like "I don't trust him not to betray us" or "I don't like it when I can't find my keys."
Thus, "Nobody who doesn’t support Trump will ever trust their judgment or patriotism again" means "Anybody who doesn’t support Trump will not ever trust their judgment or patriotism again." It's talking about people that don't support Trump, but the negative indefinite pronoun nobody is used because the sentence is describing something that people in this group will not do.
It doesn't really make sense to try to interpret "nobody who doesn’t support Trump" outside of the context of the negative clause. The word nobody doesn't actually refer to any existing group of people: compare the erroneous formulation of (joke) arguments like "a cheese sandwich is better than nothing, and nothing is better than God, so a cheese sandwich is better than God". This argument is flawed because a sentence like "Nothing is better than God" doesn't actually mean "There exists something called 'nothing' that is better than God"; it means "There does not exist anything that is better than God."