Glaube nicht!

Ditto. cf Google.

Newly-diagnosed does not make sense.

Either the illness (or whatever) was re-diagnosed (either the diagnosis was changed or the same illness was found a second time), or it was recently diagnosed, or the act of diagnosing was repeated (because, for example, the first time the results were unclear).

For "neu-diagnostiziert" take your pick: it depends on the context. Do not use "newly diagnosed".

cyn: why does newly diagnosed not make sense?
For example, here there's an entire section for the newly diagnosed. http://www.mssociety.org.uk/

OK, I went to that site and clicked the button "Newly diagnosed" and the first thing they do is explain what they mean by that, which is "just diagnosed". They drop the "newly" right away:

Just Diagnosed Pathway

This pathway is for those who have just been diagnosed, or are in the process of being diagnosed with multiple sclerosis (MS). It is also for those who are the partner, relative or friend of someone who has received a diagnosis.

In other words, "newly" is so ambiguous, that it instantly needs explanation. If you use recently diagnosed (which is what they mean by "just diagnosed"), or re-diagnosed, or repeated diagnosis, there is no ambiguity.

Consider this: What does the word "neu" add to "diagnostiziert"? In den allermeisten Fällen würde es vollkommen genügen, "neu" völlig wegzulassen. Diagnostiziert ist diagnostiziert. In dem Falle, dass das gerade "eben" passiert ist, würde man "neulich" diagnostiziert schreiben. "Neu-diagnostiziert" ist eine Sprachfaulheit.

Looking at the "newly diagnosed" hits on www.google.co.uk shows that they refer mainly to the person diagnosed: eg "Newly Diagnosed with Breast Cancer" http://breastcancer.about.com/od/newlydiagnosed . They are there specifically for people at a transition point. It seems to me just the right term to apply. Like "newly wed", etc.