What happens when two nouns are separated by "or"?

We all know that when two nouns are separated by “and,” you use a plural noun and a plural pronoun, like this:
Liz and Gayle have to decide.
But, what happens to the verb and pronoun when two nouns are separated by “or”?
If both nouns are singular, you use a singular verb, like so:
Liz or Gayle has to decide.
But what if one noun is singular and one is plural? The rule is that when two nouns are separated by “or,” the verb usually agrees with the noun closest to it.
Neither my demeanor nor my arguments were persuasive.
Neither my arguments nor my demeanor was persuasive.
Here’s a great example of a mistake.
“The authority of the school is superseded when law enforcement or the Department of Children Services become involved,” said a district spokeswoman.
Either one or the other can supersede the authority of the school; in other words, it doesn’t take both acting together (which is why it’s “or” and not “and”).
Both these nouns—law enforcement and the Department of Children Services—are singular nouns. So, the verb, which in this case is “become,” should be becomes: “when law enforcement becomes,” when the Department becomes….”
Now, we’ll note that this is a conversational use. Because it’s conversation, we’ll grant the spokeswoman some slack. If this was formal writing, however, it’d be wrong.