Modified from an earlier answer of mine

It is a truism to state that military units thrive on traditions and there is a strong memory culture within various units that helps bind recruits to the institution. Military traditions posed a particular problem for both German states' military forces in the aftermath of the Second World War. Pressure against celebrating a military past came from both within and without the two Germanys as there was a strong current of opinion that held German militarism as responsible for German aggression in both world wars. But rather than inventing new military traditions wholesale, the German military culture on both sides of the iron curtain celebrated a very selective reading of their military past that these militaries constructed to be the antithesis of the military under National Socialism.

One surprising area of common ground for both the National Volksarmee and the Bundeswehr was their celebration of both the Prussian military reformers like vom Stein, Scharnhorst, or Gneisenau and Prussia's battles against Napoleon during 1813. The celebration of these heroes and 1813 diverged quite markedly though. The GDR cast the Prussian reformers as proto-class warriors that realized that uniting the masses was now possible and had a revolutionary potential. For example, Alexander Abusch, one of the early luminaries of the GDR's Kulturbund drew a direct line between these Prussian reformers and the mass warfare practiced by Lenin and Stalin in his 1946 survey of Germany's historical maldevelopment, Der Irrweg einer Nation. Abusch contended that the reactionary Prussian government ignored the import of the reformers and the military focused only upon planning for military operations, not a mobilization of the population. The Prussian reformers also had the added benefit of fighting in a war in which they were junior partners of the Russians and thus were a prequel to the "socialist brothers in arms" that permeated propaganda about the Warsaw Pact. The NVA made both Blücher and Scharnhorst the subject of medals for valor, and the NVA enshrined these generals as archetypal practitioners of people's warfare. This selective reading of the past was in keeping with the larger Soviet wartime trend of celebrating tsarist generals like Suvorov or Bagration as historical individuals that transcended their class origins in the name of a people's patriotism. It certainly also helped the NVA that the Napoleonic Prussian military was an ally of Russia, and thus celebration of the past fit within current geopolitical realities.

The FRG's approach to the Prussian reformers differed somewhat from their Eastern counterparts. Instead of being leaders of a people in arms, instead they were cast as advocates of modern citizen-soldier. One of the architects of the FRG's rearmament, Wolf Graf von Baudissin, would in various speeches and articles draw explicit parallels between the Bundeswehr of the 1950s and those of the earlier reform era. The military traditions of the past were not passive symbols according to von Baudissin, but rather a military past repurposed for a democratic present. The decision by the Bundeswehr to use the German Iron Cross as its symbol was one that was highly contentious but justified by the FRG Defense Ministry because it was a symbolized the democratically-minded army of 1813, not the later armies of Prussia and the Third Reich. The Bundeswehr returned to the Wehrmacht's style of saluting and standing at attention, but von Baudissin insisted that a soldier should only be under the requirement to salute only his immediate superiors, lest older habits of slavish deference return. Other unit traditions became similarly enmeshed in the idea that they were in service to a democratic state that would defend Europe against totalitarianism. Tradition rooms on Bundeswehr bases would help ground new recruits in a democratic ethos by connecting them to now sanitized forebearers.

The official stance of the Bundeswehr towards Frederick the Great was pretty ambivalent. Despite Frederick's accomplishments, he still has an uncomfortable association with Prussian militarism and the Third Reich's unabashed admiration for him. When the German state reinterred his body at Potsdam in 1991, the provision of a Bundeswehr guard and Helmut Kohl's presence were highly contentious political issues.

There are two major exceptions in the official commemorations of the Frederican era. Firstly, Frederick the Great's main cavalry commander, Friedrich Wilhelm von Seydlitz, who famously won the Battle of Zorndorf by disobeying his king's orders. A number of military intellectuals in the Bundeswehr connected von Seydlitz's initiative as in keeping with the philosophical-military concept of Innere Führung, in which a subordinate was supposed to keep his head and disobey orders if the situation warranted it. Von Baudissin and other Bundeswehr reformers saw the cultivation of this initiative as an alternative to the blind obedience they blamed for the Wehrmacht's conduct in the Second World War. That von Seydlitz's initiative carried the day made him a doubly attractive figure.

The other line of tradition and continuity is military music and its affiliated ceremonies. Bundeswehr bands still frequently use the famous Prussian marches as a part of their repertoire. The military tattoo, a staple of the Frederican army, is still a popular activity of the Bundeswehr. The Wachbataillon, an elite honor guard used at state ceremonies, carries on many of the traditions and drills of the Prussian 1. Garde-Regiment zu Fuß. Beyond the quaint use of the ancien regime music, some of the Wehrmacht's songs like the Panzerlied have become enshrined in many local unit traditions. These song's clear connection to the Third Reich have made their use by soldiers and recruits a source of embarrassment for the military.

One area of its military culture in which the FRG and the GDR parted ways was their respective attitudes towards the Wehrmacht. The Bundeswehr developed an early version of the "clean Wehrmacht" and celebrated the common German soldier for doing their duty for a criminal regime. In this formulation, the common German soldier became a victim of the regime that ordered him to fight a war he could not win. Heroes of the Third Reich like Günther Lütjens, Werner Mölders, and Erwin Rommel had prominent Bundeswehr bases, units, and ships named after them on the rationale that they were martyrs of the previous regime's criminal incompetence that squandered their lives. Some of these new formations were led by former Wehrmacht officers like Gunter Rall and fed into this presentation of a sanitized military past.

In contrast, the GDR already had a large pantheon of KPD and other political martyrs to name NVA formations after. Additionally, the GDR derived a good deal of propaganda value from presenting the FRG as only superficially denazified and the presence of these veterans in the Bundeswehr added to this. The NVA did recruit from veterans as well, but a number of them came from the Nationalkomitee Freies Deutschland and Bund Deutscher Offiziere, which were organizations set up during the war for German prisoners of war to participate in in anti-Nazi activities.

Cracks in the edifice of this sanitized version of the military past began to appear by the 1960s and 70s. The NVA's Prussian-style uniforms and goosestepping became indelibly associated in the West with the Berlin Wall, not the obscure martial traditions of 1813. The Bundeswehr of the 1960s and 70s also became a haven for anti-communist sentiments and its close association with Germany's large veteran culture caused periodic embarrassment for various FRG governments. Military traditions sometimes slithered into a clandestine celebration of the military triumphs of the Third Reich. In his autobiography, the British test pilot Eric Brown recalled being invited by his German hosts to a backroom of a Bundesmarine bar where Kriegsmarine ensigns and other trophies were kept on display. One of the most famous and emblematic scandals was the Rudel Affair of 1976, where Hans-Ulrich Rudel, a famous stuka pilot and a darling of the FRG extreme right, attended a reunion of the Immelmann Squadron at the Bremgarten airbase in 1976. When questioned about it, Luftwaffe General Walter Krupinski and Karl Heinz Franke committed a faux paux when questioned about Rudel's attendance commented that Rudel had as much right to be active as former KPD members like Herbert Wehner were active in FRG politics. Comparing an active member of the SPD unfavorably to an unrepentant National Socialist like Rudel was too much for Bonn to bear, especially right before an election, and the generals were cashiered.

I'm a bit confused by the difference between the title and the body of your question. The first seems to ask if Prussian military tradition continued (presumably in the West) German military during the Cold War.

That's a difficult question to answer, since it is hard to say what "Prussian military tradition" was by the 1950s. But I can say that a number of the key people that were involved in the creation of the Bunderswehr (such as Guderian and von Manteuffel) were from the "Prussian" parts of Germany.

Further, one of the major ethos/themes of the Bunderswehr harks back to the ideas and beliefs of the Prussian military reformers of the 1800s, such as Scharnhorst.

It's a shame no-one seems to have an answer for this question, 30ish upvotes suggest it's a good question but maybe something that simply hasn't been explored.