I think Hitler will invest a lot in genetics to make "real germanies". That should not mean, that he wants to clon people.
I think he will to modify the DNA of the babies so that they have blue eyes and blond hairs. ... just German

Well, the Earthlings probably have taken the genmods on to new height. Think of today's plastic surgeries, boob-jobs, penis enlargements, etc. The brave new world of 2018 probably has their own genmod doctors who will make your body parts bigger, better, and more beautiful, just that what the exact new "fad" or "fashion" is is debatable.

Glossary:
genmod = genetically modified body part

Now, for A. Huxley (same initials as A.Hitler :) ... I have not read Huxley's books, but I recall reading some marketing abstract on the back of the book. I recall his future had leaders genetically engineering servants who were "happy" doing the manual labor they were "designed" for. This is modifications to their "genetic hard-wiring of the brain" if you can pin-point such a thing. Or perhaps it was about brain-washing them since childhood for some task. I think the brain-washing part could go with the Moon-Nazis quite easily.

Bored of cloning. Why is cloning the subject of every damn scifi concept? zzzzzz

I agree that it kind of connects to the Nazis desire to breed the perfect Aryan, but the Nazis were also big romantics and that shouldn't be forgotten. In other words, they had this glorified fantasy world vision of the "master race" who were NATURALLY superior (ie: they wouldn't need any kind of scientific enhancement) to other supposed "races".

In reality, there is no such thing as different types of human "race". We are all the same. But are the Nazis still living in their deluded fantasy? Presumably. I think the idea of cloning the 'master race' would have been distasteful to them. Clone some subhuman slaves, maybe.

The Nazis were big naturists and were ahead of their time in terms of protecting the environment (didn't they make laws protecting their forests and such like?) How would the Moon Nazis deal with the lack of nature on the moon - build big artificial forests?

As far as the effect of reduced gravity on the bodies of generations of Nazis brought up on the moon - I like to imagine they would come up with monstrous, industrial-looking machines to deal with such things (visually, it strikes builds on the theme of "Iron Sky").

How about if everyone sleeps (or spends part of their day in) in a big centrifuge?

Good thinking, but you're forgetting one thing. The Nazis were quite fond of experimenting extensively, and preferably not on their own (but extreme circumstances call for extreme measures). As we all know, thousands and thousands of people were killed this way. But how many Nazis actually took off to the moon? Couldn't have been more than a handful, maybe a few thousand or so. That wouldn't have left them much to experiment on. They would have taken great care with every subject, of course, but still - could they have come that far with so few test subjects (and so few means)?

One way to solve this would be to have them create clones - lots and lots of them, expendable ones, like in the 'Star Wars' prequels.

their "radical" researchmethods would have resulted in a more advanced medical knowlege. mostly in injurytreatment and pysical developement.
living on the moon vor seven decades with 1/6 of the usual gravitation would leave the current generation of nazi-soldiers high grown weaklings with glassy bones without any stamina. prakticably absolut unfit to battle earthsodliers on equal terms.
so the moonnazis would have to counteract the effects of the low gravity, maybe even turn that in an advantage.
i think artificial gravity is kind of far off.
so they would have genetic threatments and drugs, special trainingequipment to keep their bones and muskels on earthlevel or even push them above. in addition they could reduce the airpressure in their moonbases to simulate an high altitude training (and to preserve energie)
thinking of our ability to provide artificial replacement vor nearly every major bone in the human body, i would think it is not far off to think moonnazis would have even gone to the next level.
without ethical resentments they could have found a way to solve the problem that come with implementing foreign objekts into the organism. i'm not talking about startrek-borg but something like an eary forecast
(last moment of of the movie- a beatup-nazi-saucer taking of to the deltaquadrant ... just kidding ;) )

The Nazis certainly were interested in genetics. Good point.