First, none can be used in 2 contexts, singular or plural; it can mean all of them or no one or not a single one. In your sentence "None of them have told me". means all of them did not tell you, and if you use another way to state it, this way; "None of them has told me". means not any individual or not a single one of those people.
Second, if either and neither are used as subjects of a sentence then, the verb should be singular. So, it is correct to state, "Neither of them has told me".
Some more examples; "Neither of the applicants is qualified". "Neither of the students stands out". But if neither nor is used, it means that it is being used as correlative conjunctions and the verb should agree with the subject closest to it. Examples: "Neither the students nor the teacher is given the chance to explain". But other experts say that if a sentence contains both singular and plural subjects connected by either/or or neither nor, put the plural subject second or last and the verb should be are, were, and have, simple present or simple past as the case may be. The past of have as well all know is had. Correcting the sentence then according to the rule, we state; "Neither the teacher nor the students are given the chance to explain".
Sentence number one is correct. There is no problem at all with treating "none" as a plural. In fact, it is by far more logical to do so, especially when the verb follows a plural pronoun - even if that pronoun is an object pronoun. I feel that it is purely subjective, inflexible, and pedantic to say that treating "none" as a plural is incorrect.
I would likely say "neither of them has". However, I wouldn't object to "neither of them have".
I'll post from references books later for those who argue against what I say, though it's not necessary. What I say is correct.
You'll find that "none" is treated strictly as a singular pronoun in edited writing. In my opinion, native speakers who "learn" the "correct way" will likely maintain that "none" must always be singular as well. It's very natural to treat "none" as a plural. It makes sense to do so, and it is logical.
You might find in a grammar book that the explanation for treating "none" as a singular is that this is "formal usage", and that treating "none" as a plural is "informal usage". However, in my opinion, this explanation is nothing more than wriggling out of taking a genuine stand on what one really believes and knows to be the truth in order to conform, not be criticized, and to maintain an artificial sense of propriety in the interest of selling books and not being fined by the grammar police.
You probably know that verbs are inflected for person and number. Take the verb 'have'.
He [ 3rd Person, singular ] has
They [ 3rd Person, plural ] have
So, all you have to decide is: How many people does the pronoun 'none' represent? One, or more? You see the difficulty now? 'None' is no people!
Verbs and pronouns must correspond. You would not write: '*He have', because he is singular. Nor would you write '*they has'. This problem does not arise with none and neither. You can write has or have. No one can say you are wrong, as the feature singular or plural is not expressed on them.