The doubts started to appear among the Wehrmacht soldiers in winter 1942/43 (Battle of Stalingrad). By summer of 1944 the skepticism about the chances of Germany winning the war became widespread.

Gunther K. Koschorrek wrote in his book, "Blood Red Snow. The Memoirs of a German Soldier on the Eastern Front":

Wounded in December 1942, the author was put on a hospital train together with several soldiers flown out of the encircled 6th Army at Stalingrad.

"One soldier with a bandage round his head, and who only can see out of one eye, ridicules the latest Army radio report, which plays down the disaster in Stalingrad and refers to the defeat of the Sixth Army in a highly stylised propaganda report about the willingness of the German soldier to make a heroic stand."

Still, the author noted that the few acquaintances back home in Insterbrug, who questioned him about the Eastern Front, did not believe the information about the dampening morale of the German troops there.

The other memoir, "A Mind in Prison" by Bruno Manz, stated that in June 1944 the public was already losing faith in the Nazi leadership:

The author was on a home leave in June 1944. He served in Finland, which was a relatively quiet area compared to the the Eastern front.

"As I was reentering my railcar, I overheard the conversation of two officers. They used an expression I had never heard before, but would hear with increasing frequency in the months ahead, particularly after the war. It was Gröfaz, a sardonic acronym for Grösser Feldherr aller Zeiten (greatest military genius of all times). At the time I did not understand its meaning, much less its ridicule, but I had the uneasy feeling that it was an unflattering nickname for Hitler. After the war I learned that it was used mainly by disillusioned officers and other people, who as far as information was concerned, were more privileged than I was."

In the same chapter the author noted that his civilian father, an eager Nazi, was depressed about the war developments, but still hoped for the situation to reverse itself.

Based on this and other wartime memoirs and personal accounts, it seems that doubts about the war outcome were likely to start developing after the Battle of Stalingrad. When Allies landed in Normandy, the disappointment with Hitler's leadership became common.

The citizens as seen above realized the war is over quite late due to Goebbels doing a great work in propagating the war.

Most of the military commanders realized the war is lost in 1943. due to Soviets turning the tide of the war at Stalingrad and Kursk and Afrika Korps losing the battle in North Africa.

All of this aside im sure there were plenty of fanatics who believed the war is winnable till the last moment but these people were mostly members of the SS or just pure fanatics.