Hey there! You might be interested in this thread from a couple of months ago which covered a similar topic. The tl;dr is: Yes, the West German Bundeswehr (and by extension NATO itself) did get her hands on the Soviet-built military hardware of the East German forces, as they were incorporated into the unified German military.
Many pieces were still used for quite some time for exercise purposes (small arms, APCs, tanks) but subsequently sold to other countries (mainly Turkey, Greece, Indonesia). The aircraft however proved to be the most interesting to western analysts as the capabilities of the MiG-29, the latest Soviet fighter jet (of which the GDR maintained 24), were unknown in detail at the time. In 2004 the last 22 MiG-29 were sold to Poland at a symbolic price.
Surprisingly, the Soviets allowed (to my knowledge) the GDR to keep all military hardware they had purchased from the SU even after the fall of the Berlin Wall and the eventual reunification with West Germany. Obviously any military hardware that was deployed directly with Soviet Union forces, which were not made available for easy examination, were removed from the GDR shortly after the fall of the wall.
While this was certainly an opportunity for Western Intelligence services to begin examining military equipment previously unavailable to them (and in many cases, much of the equipment the GDR possessed wasn't unknown and was likely previously examined due to arrangements with other countries that had purchased Soviet hardware), NATO didn't really start fully examining some of the Soviet Bloc equipment until a few years later.
For example, the GDR had purchased shortly before the fall of the wall and reunification, 24 MiG-29 Fulcrums. NATO and Western intelligence did examine these as soon as they were given the opportunity to do so, as they were relatively new 4th generation fighters, and had an array of technological advancements not seen on previous Soviet fighters. Eventually, these were made NATO capable, intergrated into the Luftwaffe, and later used as evaluation aircraft in order to test the capabilities of NATO aircraft against Soviet aircraft and vice versa.