As far as I am aware, the only weapon that made it into service with the Bundeswehr that has it's direct origins in the Wehrmacht was the MG3, a direct derivative of the MG42 which entered service in the late 1950s. When the Bundeswehr was formed, their equipment was American supplied, with everything from tanks to combat aircraft being surplus from the American armed forces.

In terms of people, much of the original organization of the Bundeswehr was carried out by former Wehrmacht officers. Although those with close Nazi ties were obviously precluded in theory, that didn't preclude some pretty senior men from the Wehrmacht from participating such as Hasso von Manteuffel. In general, if you wanted to create a competent fighting force, it would be foolish not to use some of the battle hardened veterans of the Wehrmacht when doing so. However, the Bundestag was very careful to screen whom they appointed to command positions. Indeed, most of the surviving Luftwaffe aces joined the new air force as well.

As for bureaucracy, it's important to note that the bureaucracy of the Wehrmacht was pretty...confusing, particularly as the war wore on. The OKW and OKH rivalry was really indicative of the types of infighting that occured within the armed forces. Much of this was remnants of the Prussian officer class and its clashes with new blood the entered the armed forces in the lead up and during WW2. One of the symptoms of this that actual functional state control over the Wehrmacht was severely limited. In contrast, the Bundeswehr was by design subservient to the Bundestag, clearly a design to avoid giving back a voice and influence to the military.