I do not understand the answer about firmware sending unicode characters. The keyboard does not send the characters, but only the key position. Then the OS comes into play. And Volker, I am sorry to say, there is no simple solution as soon as you go away from just using a US English layout. You will sooner or later find situations where the keyboard will not trigger what you want it to do. That is because how the OS interacts with the keyboard.
On Windows the scancode (which basically is the key position on a standard ANSI, ISO or JIS keyboard) is linked to a keyboard layout which you choose as a language. In the keyboard layout itself there are two definitions. That is first a virtual keycode. You find that in documentation often as VK_… And a key also is assigned a unicode character. But those are two definitions made in the keyboard layout file. Depending if a program queries the scan code, the VK or the character it can behave sometimes in ways which are unexpected. That is because many (most) programs assume you are using a US layout.
What is the best solution to realize your NEO-QWERTZ layout? That depends on the access you have on the PCs you are using. You will get the least problems when you choose a language / keyboard layout which implements your wanted behavior. I am not sure if there is a layout you can download for NEO-QWERTZ, but think so. Otherwise one could create one with kbdedit (paid software, but not too expensive), or with some limitations also with Microsoft Keyboard Layout Creator. To install the layout you will need admin rights.
If you can not change the keyboard layout on your PC there will be no way from the UHK alone to directly give you the greek characters and so on. You can partially work-around with the macros although, with the limitations mentioned.
Another option is to use a software to remap your layers in the way you want. Kanata works great for that. But here you will also find some niggles, for example auto-repeating might not be working and things like that.
Finally, if you are happy with the Neo layer concept that is great. I personally find it much better to create your custom layers – which use layer switching keys which are easier to reach than what Neo does (also does not work well with an ANSI-keyboard).
You might find my article about DeutschlandPlus interesting. I also have published three newer articles with lots of thoughts about creating or finding your personal best keyboard / layout solution.
My articles are on kbd.news:
In summary: for a few characters and symbols I would introduce a layer to be accessed with the CapsLock key (position) as the layer switch. Add a navigation layer with the held space key (SpaceFN concept) and you should be happy. If you need a few symbols like greek characters and so on, use a text expansion program and define what you need. Way more flexible and easier to type. For the latter I can recommend Espanso.