My dad is a former Cold War soldier, and he told me that the Canadian military (as with all the other allied nations) were there at the request of the Federal Republic of Germany (also known as West Germany), and if they wanted the Germans could have told them to leave. But of course, why would they.
So anyway, at what point did West Germany transfer from an area of occupation post-WW2 to a more autonomous state? My dad served in Germany from 1982 to 1985, so this was really late cold war. Other than that timeframe, I know that the Saar Protectorate (which was under the control of France) was returned to West Germany in 1957.
Generally speaking, 1955 was the period in which the FRG regained most metrics of full sovereignty. The London and Paris Conferences of the prior year allowed for the formal ending of the occupation and paved the way for FRG entry into NATO. The 1955 accords were the capstone for earlier bilateral and multilateral agreements that granted the FRG a degree of semi-sovereignty and treated the Bonn government as a sovereign German nation (the position of Bonn was it was the sole German nation too, and the broader Western reaction of what became known as the Hallstein Doctrine was more complex) since 1948.
The two major areas which Bonn did not have control over in 1955 were the status of West Berlin and the ability to initiate a final peace treaty that formally ended the Second World War. Both of those areas were the preserve of the big four Allied powers and the Cold War precluded any substantial headway in resolving these issues. Bonn could not, for example, give West Berlin to the GDR without the Allies' permission as this would have been a violation of the four-power agreements on Berlin. This is one of the reasons why West Berlin's governance was quite wonky during the Cold War; its residents were FRG citizens, but its federal representatives were non-voting members of the Bundestag. Military government remained much more visible in West Berlin as memories of the wider military occupation faded in the rest of the FRG. While the nascent Bundeswehr soon took up responsibility for defending the inner-German border along with its NATO allies, it could not legally maintain a presence within the divided city.
Sovereignty and independence though are more than just paper declarations. One of the dominant features of the FRG's intra-European diplomacy in this period was that it had to navigate within networks and systems predicated on reducing the amount of latitude Germany had within these alliances. This imparted a degree of friction into diplomacy in this period, especially when memories of the war were still fresh. Much of the debates around German rearmament revolved around how to ensure that the FRG could use its resources to defend Western Europe without becoming too powerful. The initial impulse among non-German diplomats was to foster the European Defense Community (EDC) which would create a unified pan-European army and leadership. Not only would this ease equipment problems, it would subordinate a German army to non-Germans. For its part Bonn caused enough ripples in the EDC negotiations by promising to meet or even exceed its commitment leading the French parliament to scuttle the EDC, forcing diplomats to come up with a solution. The fallback to the EDC was a somewhat moribund treaty, NATO, and it allowed for the creation of a separate German national army. This solution was arguably exactly what Adenauer wanted as it maximized FRG sovereignty and treated the FRG as a normal sovereign state.