An interesting question! It touches two important fields: What was the legal framework for displaying WWII medals, and what was society's attitude towards them? I didn't have time to answer until now, and I had hoped someone more qualified would pick it up. But since nobody else did, here goes.

First of all, I will only answer this question within the scope of West Germany, because I just don't know enough about any of the other regions. This would be very interesting to see a more comprehensive answer to. My gut feeling (dangerous, I know) is that there are a few very different "classes" of answers regarding the countries: (1) West Germany; (2) East Germany; (3) Eastern Block (ok, non-Western, so it also covers the Croatian Ustaša, but especially Hungary and Romania); (4) Italy; (5) Japan. But, as I said, back to West Germany.

So the situation in West Germany is this: the war is over; the country almost completely destroyed, at least the urban areas; refugees (mostly) from the East swarming into areas that are already strapped for food and shelter. And on top of that, the war ended disastrously, both politically and morally. Politically, it ended with an unconditional surrender, the dismantling of the government, and the partition of the country; even after the end of the occupation, the partition into West Germany, East Germany, and Eastern areas under Polish and Soviet rule (or even Czech, depending on where you came from and what your view point was at that time) was a central political topic for years to come. Morally, the war had helped spread and prolong a genocide on unparalleled scale.

The situation is thus quite ambiguous. On the one hand, the population as a whole did not want to remember the war they had just survived. There was a strong emphasis on looking forward and rebuilding the country. On the other hand, you had the war veterans who had often gone through incredible personal hardships fighting for an ideology they believed in, or at least saw as the lesser of two evils compared to the "bolshevism from the east", and who ended with naught for it, or worse than naught: having to come to terms with the idea that they served a murderous and criminal regime. Was there no point to all their hardships? How do you deal with that? For the most part, by draping a veil of silence over recent history and focusing on the future. This conveniently allows everybody not to reflect too much on their personal role, a behavior you can see in many societies that try to deal with a traumatic situation where former victims and their tormentors have to live side by side again out of necessity. In addition, the, we can assume, almost always traumatic experiences of such a total war made it hard for many to reflect on them, and instead they tried to forget them. Anecdotally, neither of my grandfathers ever spoke about their war experiences, and this seems to be far from uncommon. Those types of persons naturally had not much of an urge to present their medals. Also, in a society were most male citizens had been conscripted, the idea of a "veteran" as something special quickly disappears.

Of course, this doesn't hold for everybody. Those with, for example, family histories of being soldiers, or less innocuously, those who still were attached to National Socialism, might have had a desire to present their war medals. [1] lists several dozen incidents of meetings, association foundings, press conferences of such associations, and protests against their meetings, for the first decade of the FDR (1949-1959). This includes such groups as the "Verband deutscher Soldaten" (association of German soldiers), whose first documented press conference after its founding was a condemnation of the July 20 plot, and the reconstitution of the "Stahlhelm", who would go on to elect Albert Kesselring, a German marshal with a questionable track record and who was still in prison, as its chairman. Kesselring managed to become somewhat of a cause célèbre for postwar West German Vergangenheitsbewältigung, or the lack of it; both the public and the press had lobbied for his release, which eventually was granted. Of course, for the same time frame, [1] list much more large protests against rearmament in Germany, and later against atomic armament, as well as first steps towards reconciliation, while there were still the occasional desecrations of Jewish cemeteries, or damaging of memorial plaques. So, all in all, a society not all that much unlike many western societies today (were stuff like that, sadly, still happens), though maybe more fundamentally divided, and with unstable foundations.

Which finally brings me to the second part. If I wanted to wear my medals which I had gotten during the war, was I allowed to? The answer is a resounding "it depends". First of all, the Nazi Government created a plethora of medals for different purposes, sometimes with a number of classes, inflating as the war went on. Some of them could be clearly seen as "military" (the Iron Cross, for example, though even those might have on occasion been awarded for morally questionable conduct), others clearly as "political" (for example, the Blood Order for early membership in the party and strong adherence to its ideology); but this was not always as clear-cut. Second, the Nazi Government had a tendency to plaster swastikas everywhere, even on purely military medals; this was problematic because the swastika was a forbidden propaganda symbol which could only be publicly displayed in very limited contexts. The solution in the end was a law with an appendix showing "de-nazified", authorized versions of those medals that were allowed to be displayed. Everything else was forbidden. You can see the appendix here. For example, on pages 6 and 7, you can see the new versions of the Iron Cross and the German Cross. For comparison, wikipedia shows the original versions with swastikas here and here.

[1] Wolfgang Kraushaar, Die Protest-Chronik. Rogner & Bernhard, 1996