So it's possible that Germany would have protested against nukes deployed on their soil. Would the Green Party have more popularity because of this and would the military have had as much funding?
Let's not put the cart before the horses here. Before we address the Greens' founding in 1980, we need to address the 1957 elections and the
Kampf dem Atomtod movement.
What happened IOTL is that Adenauer and Strauß, who were strongly pro-nukes, ran a double strategy. On the one hand, they publicly supported US nuke deployment on FRG soil. On the other hand, they made secret plans for a joint nuclear program with France and Italy. While the latter plans remained secret until after they had fallen apart with de Gaulle's 1958 coup, Adenauer's and Strauß's support of deployment of US nukes became public knowledge in early 1957. In April 1957, Adenauer famously downplayed tactical nukes as "advanced artillery".
CDU/CSU won a majority in the 1957 elections; the economy was doing great and while the West German people wasn't exactly happy about German nukes, the opposition's attempts to exploit the discontent failed. Actually, the 1957 elections were the only time a list won a majority in the FRG.
The opposition parties, SPD and the newly left-leaning FDP, being out of parliamentary options, joined forces with labour unions and liberal clergy members to create Kampf dem Atomtod, an extra-parliamentary movement dedicated to stopping deployment of nukes on German soil. The movement had huge support, with the vast majority of FRG citizens opposing nuke deployment on German soil and 1,5 million Germans joining the KdA protests in early 1958.
Kampf dem Atomtod ultimately failed because they could not agree on tactics. SPD and labor unions tried to organize a plebiscite, which was forbidden by the Verfassungsgericht as the FRG constitution doesn't have a framework for referenda. More left-leaning KdA members favoured a general strike strategy, which was opposed by SPD and DGB and therefore did not come to pass (but might have succeeded despite its illegality, as polling showed a narrow majority of West Germans in support of a strike strategy).
ITTL, this may go into different directions. Adenauer may lose in 1957, or win a plurality and have to give up nukes. Gesamtdeutsche Volkspartei, Heinemann's short-lived pacifist center-right party, may survive until the 1957 elections, taking votes from both CDU/CSU and SPD. Or Adenauer still wins in 1957, Kampf dem Atomtod gets stronger than IOTL, an illegal general strike happens, and shit hits the fan.
Edit: Another possibility is that Adenauer fails to unite the Right, either due to a stronger revanchist vote disliking Adenauer's pro-NATO foreign policy, or because the rightist refugee party GB/BHE wins the Hibakusha vote in 1957 and makes the 5% cutoff.