Expanded from an earlier answer of mine

For the most part, the officer corps of both the Bundeswehr and the Nationale Volksarmee (NVA) drew from former Wehrmacht officers when it came to staffing their establishments. Although the size of both militaries was considerably smaller than its wartime predecessor, these veterans possessed both experience and authority that would be necessary for the tasks that could face a military within the Cold War.

Within the FRG, most senior Wehrmacht veterans that went on to serve in the Bundeswehr has to be cleared through the Personalgutachterausschuss (PGA), a committee set up in 1955 for vetting these applicants for a correct political mentality. Lower-level officer candidates were to be cleared by the Annahmeorganization (Accepting Organization), which had much less stringent criteria for denazification.

But much of the groundwork for recruiting these officers had been laid in 1950 by the innocuous-sounding Zentrale für Heimatdienst (ZfH- Center for Heimat Service) under the leadership of Gerhard von Schwerin, a veteran of the Panzer corps. Adenauaer tasked von Schwerin to prepare the blueprints for a swift FRG rearmament and the ZfH began to sound out prospective officers trough the various veterans' organizations. As this was a politically sensitive topic, these preparations violated Allied disarmament regulations and rearmament was not terribly popular within the FRG electorate, von Schwerin was tasked to proceed with great discretion. In this task he failed, and coupled with his close association with the prior regime, Adenauer found grounds to fire him and replace him with Theodor Blank in October 1950. Although Blank was a Wehrmacht veteran, he was a low-level officer and was not a career soldier. The now renamed Amt-Blank worked closely with German generals like Hans Speidel (Rommel's chief of staff) and Adolf Heusinger to lay the foundations for a German rearmament. As Adenauer navigated the political rocks and shoals of getting European acceptance of FRG rearmament, first through the European Defense Community (EDC) and then through NATO when the former collapsed, Amt Blank silently prepared for a new German military and doctrine.

The GDR went through much a similar process of covert process of preparation for rearmament and then a public announcement of a large defense establishment. The NVA grew out of the various paramilitary police and border patrol formations that the Soviets had created at the end of the 1940s. Like their Western counterparts, the Volkspolizei and later the NVA set up a screening process for vetting officers to ensure that no NSDAP ideologues made their way into the new military. The Eastern screening process departed from the FRG's in two very important ways though. Firstly, it operated through existing structures for Soviet reeducation and indoctrination of German POWs. The Soviets placed great stock in the conversion of POWs to communism, both to prevent unreformed fascists from reentering the German body politic and to present such conversions as a public propaganda coup both to their fellow Germans and abroad. A large number of the NVA's initial drew from former POWs captured by the Soviets; soldiers never captured or captured by the Western powers were too unreliable. Many of the Volkspolizei and NVA's flag officers were brought directly back from Soviet captivity to once again wear the uniform. Secondly, while the Bundeswehr prided itself in the creation of an apolitical German soldier, the NVA's officer ranks were to be composed of explicitly SED party loyalists. Many Wehrmacht veterans found it very difficult for the SED state to accept their claims of loyalty. This meant that after the initial establishment of the NVA, the SED quickly replaced these veterans with its own cadres in the early 1960s, whereas in the FRG, Wehrmacht veterans in the Bundeswehr became part of the FRG political establishment.

The entrenched position of these veterans within the Bundeswehr was a source of continual embarrassment for Bonn during the Cold War. PGA screening was much like the rest of the denazification procedures in that it was not very effective. Much of the cooperation Adenauer received from the veterans involved in Amt Blank was predicated on Bonn pursuing the issue of clemency for convicted war criminals in Western Allied jails, an unspoken quid-quo-pro that Adenauer honored and the Americans agreed to. The Bundeswehr and the FRG veteran community in general became bastions of anticommunism and their Cold War rhetoric at times verged on wartime visions of barbaric hordes of Slavs ready to conquer the West. The Bundeswehr also enshrined the notion of a clean Wehrmacht that fought its war honorably and was unrelated to Nazi aggression. While there were some unrepentant Nazi veterans who refused to heed the colors, these individuals were active in various veteran communities as well as neo-Nazi organizations such as the Deutsche Reichspartei. Bundeswehr officers often kept a very chummy relationship with these types and this created a number of scandals. Bundesmarine officers, for example, showed up in full uniform at Karl Dönitz's funeral despite being ordered not to do so. More famously, the so-called Rudel-Affäre (Rudel Scandal) led to the resignations of Luftwaffe generals Walter Krupinski and Karl Heinz Franke when they praised Hans-Ulrich Rudel at a reunion on a Bundeswehr base. Not only did they invite Rudel to the reunion, they praised the openly-unrepentant Nazi Rudel as a decent soldier and compared him unfavorably to the SPD politician Herbert Wehner who fought Nazism in exile during the war. The collective weight of these scandals were a blight on the Bundeswehr and something that the GDR's media was more than willing to publicize. But Bonn tended to be reactive rather than proactive over the presence of Wehrmacht veterans with questionable politics in the Bundeswehr and these scandals only ended when this cohort either died or retired.

Sources

Abenheim, Donald. Reforging the Iron Cross: The Search for Tradition in the West German Armed Forces. Princeton, N.J.: Princeton University Press, 1988.

Grift, Liesbeth van de. Securing the Communist State: The Reconstruction of Coercive Institutions in the Soviet Zone of Germany and Romania, 1944-1948. Lanham, Md: Lexington Books, 2012.

Large, David Clay. Germans to the Front West German Rearmament in the Adenauer Era. Chapel Hill: University of North Carolina Press, 1996.

Lockenour, Jay. Soldiers As Citizens: Former Wehrmacht Officers in the Federal Republic of Germany, 1945 - 1955. Lincoln: Univ. of Nebraska Press, 2001.