East Germany, formally the German Democratic Republic or DDR in its German acronym, existed for 40 years, 11 months, and 4 weeks. The brief lifespan of East Germany’s military, the National Volksarmee, is well-documented and studied by military historians. It only used a limited amount of ex-WWII kit for a short while.
However the nation had paramilitary forces which did use WWII weapons for a longer time: the Grenztruppen (border troops), the Volkspolizei (national-level police), and the KdA (a second-rate militia). They have received less historical attention.
(Officer of the East German Volkspolizei with a WWII German StG-44 assault rifle.)
(East German Grenztruppen with WWII Soviet PPSh-41 submachine guns during the 1960s.)