To be clear, Germany today does not own or control by themselves, any nuclear weapons.

For the purposes of the 20 year rule, though in essence it is unchanged today, what they had access to through the end of the Cold War was the sharing of nuclear weapons under NATO from the US.

Essentially the US would place selected nuclear weapons at a joint base in the host nation, typically B61 bomb, a "dumb" bomb capable of being delivered from a whole host of aircraft. These weapons would be stored at a base in the host country with USAF personnel guarding and maintaining them. Then should the need arise, specially trained and compatible aircraft of the host nation would be loaded with the bomb and included in NATO nuclear strike packages, while the US controlled the Permissive Action Links, basically the codes which actually armed the bomb. In days past other nuclear weapons such as short ranged missiles, or even artillery shells would also be part of the equation, however these have for the most part been phased out.

There is meanwhile a significant number of TV-documentaries and authoritative articles to this topic, especially in Germany (here a very good article, providing a nice summary: Als die Atombomben-Träume platzten).

At the time the Bundeswehr (German military) was re-established, in mid-1950s, German politicians were gravely concerned about a possible Soviet attack on West Germany - but also about the 'nuclear monopoly' of the USA and the USSR.

Correspondingly, (West) German Chancelor Konrad Adenauer followed what could be described as a three-prong politics: he entered negotiations with France for joint development of nuclear weapons; he decided to launch development of nuclear weapons at home; and he requested the deployment of nuclear-armed US units to bolster defences of his country (and this request was granted in form of artillery units equipped with long-range howitzers that could fire nuclear shells).

Development of nuclear weapons at home was stopped amid massive public protest. Work on a joint, Franco-German bomb was indirectly decided by the outcome of the West German contest for a new fighter-bomber: while the Bundeswehr (i.e. the Luftwaffe) preferred the French-made Mirage III interceptor (and fighter-bomber), Washington offered 'carrier systems' for nuclear weapons to West Germany - in form of Lockheed F-104 Starfighters, armed with US-made B61 nuclear bombs, which remained under US control (and would remain under US control until a definite order would have been issued for their deployment). In that fashion, not only the costs for deployment of US nuclear weapons in West Germany were kept lower (at least for the USA), but also Germans have got what they (i.e. their politicians) wanted: nuclear weapons.