I formerly thought of the Overman as the man who could endure and affirm nihilism. But this means that an Overman can never work toward the production and preparation of Overmen: for that would mean to politicise (illiberalise) nihilism, which is not nihilist.

Nietzsche explicitly warns that his philosophy is beyond good and evil, but not beyond good and bad. As Peter Berkowitz, whom I suspect of being a Straussian, wrote;

"[A]longside and in constant tension with Nietzsche's weighty cluster of opinions affirming that the world lacks a natural, rational, or divine order, that morality is artifice and pathology, and that the will is sovereign, exists a rival and equally weighty cluster of his opinions asserting that the cosmos has an intelligible character, that there is a suprahistorical ethical order, and that knowledge of these matters brings health, liberates, and ennobles. It is the unresolved antagonism between these sets of fundamental convictions that animates and orders Nietzsche's thought."
[Berkowitz, Nietzsche: The Ethics of an Immoralist, page 26.]

As you can tell from the title of his book, Berkowitz distinguishes between moral and ethical. He does so for the sake of clarity:

"Nietzsche's "antimoral propensity" [...] is rooted in a counter-morality, an opposing ethic, an alternative conception of what is good, right, and fitting for a human being. Thus, his criticism of morality is in fact ultimately moral or, to avoid confusion, ethical."
[ibid., page 48.]

"Good and evil" are moral values; "good and bad" are ethical values. Thus I pride myself on having resolved the "unresolved antagonism" Berkowitz believes to discern in Nietzsche's thought. As I wrote almost a year ago;

The resolution of this antagonism is the following. The intelligible - though not provable - character of the cosmos is that it is "to all eternity chaos" [La Gaya Scienza, section 109] - including the original meaning of "chaos", namely "void" - it is to all eternity devoid of meaning. Nietzsche's suprahistorical ethical order is based precisely on this insight: the measure of the rank of individual human beings as well as families, tribes, peoples, nations, and ages, is their truthfulness in this regard - how much of this deadly truth they can endure.
[...]
Thus the paradox is this: there is a natural order, one's rank in which is measured by the degree to which one can concede the absence of a natural order! It is here that we must try to understand the meaning of the subtitle of Berkowitz' book, "The Ethics of an Immoralist": there is a natural ethical order, which is based on the realisation that there is no natural moral order.

I had not completely forgotten this in reading Neumann's book, but I had dismissed it, because it rests on a valuation which true nihilism would dismiss as being merely subjective. This valuation is what I regard as Nietzsche's deepest conviction (his will, if you will). It is the last link in the following chain of thought:

Nihilism is true -> truth is terrible -> truthfulness is courageous -> courage betrays strength -> strength is good (as opposed to bad).

That strength or might is good is Nietzsche's fundamental conviction, in my opinion. His argument might be that it feels good. As he says in The Antichrist(ian);

"What is good? — Everything that heightens the feeling of power, the will to power, power itself, in man.
What is bad? — Everything that is born of weakness."
[AC 2.]

So Nietzsche is not a nihilist per se; he is a moral nihilist, but not an "ethical" nihilist, so to say.

Now as for what this was all an introduction to.

In his first 1971 lecture on BGE, Strauss said:

"The conservatives stood for throne and order, and the liberals stood for democracy, or something similar to democracy, and religion as a strictly private affair. But liberalism was already outflanked by the extreme revolutionaries, socialists, communists, anarchists, and atheists. There was a position we may call political atheism.

"Now Nietzsche opposed both the moderate and the extreme left, but he saw that conservatism had no future, that its fighting was a real garbage, and its conservatism was being eroded evermore. The consequence of this was that Nietzsche pointed to something which we may call the revolutionary right, an atheism of the right."

I must admit that I haven't read this lecture much further, and instead began reading his 1959 lectures on Zarathustra. But this insight has "illumined" me as follows. I reasoned: Conservatism meant both aristocratic and Christian conservatism. There existed a self-contradiction at the heart of conservatism, indeed, at the heart of the aristocracy. For the Christian values were the moral values "good and evil"; whereas the aristocratic values were the "ethical" values "good and bad". "Good and bad" are "noble" values, sprung from what Nietzsche called "master morality"; whereas "good and evil" are "base" values, sprung from what he called "slave morality". So why did the aristocracy champion both noble and base values? Because it had in the past divined a means to power in slave morality.

On at least one occasion, Nietzsche says that the development of something from a means into an end signifies decadence. Perhaps it signifies an impoverishment of the nobility that it got its natural, noble values so tangled up with the base values of slave morality. In any case, this self-contradiction at the heart of conservatism weakened it. By pointing to an atheism of the right, Nietzsche sought to create the possibility of a restoration of noble values. For "atheism" to him meant disbelief in the Christian god, that is, in the moral god. He did emphatically not deny the possibility of a god, or gods, beyond good and evil. So the Overman is not the man who can endure and affirm nihilism per se, that is, a man beyond both good and evil and good and bad. The Overman is the man who can endure and affirm moral nihilism. Thus to work toward the production of Overmen is not to politicise nihilism; it is to politicise immorality, to ethicise immorality.