The history of HYDRA and the Nazi Party in the MCU does not conform to the history of the war seen elsewhere. We are led to believe that, unless stated otherwise, historical events in the MCU align more or less with actual history in our universe. Just to cover the bases, though, I'll outline what happened in the comics:
Earth-616 (Main Continuity)
In the main Marvel continuity, HYDRA is a modern terrorist organization that did not exist during WWII. After the fall of the Nazi Party at the end of the war, and Hitler's death at the hands of the original Human Torch (more on that later), several of the party's top members went into hiding from Allied forces. One of these men, Baron Wolfgang von Strucker, later formed HYDRA using his own personal funds. The organization was loosely based on a secret cabal that had existed in the ancient world but disappeared during the Rennaissance Period.
Strucker and his allies - some of who were also involved with The Hand in Japan - saw HYDRA as a counter-balance to the growing popularity of capitalism & democracy in the world. They combined their belief in Facism, Socialism, and other systems to form what would become a constant thorn in the world of superheroes - HYDRA and its various offshoots.
In this continuity, Hitler was physically killed at the end of WWII but his consciousness was saved. That consciousness was later placed into clones of his original body, and he was resurrected as the Marvel villain Hate-Monger.
Earth-1610 (Ultimate Continuity)
The other continuity which the MCU draws heavily upon is the Ultimate one. In the Ultimate continuity, HYDRA is a modern terrorist organization with nowhere near the influence, resources, or power seen in either the 616 or MCU continuities. However, HYDRA in the MCU is largely based on a secret faction of Nazis in the Ultimate universe who were, in fact, Chitauri (the Ultimate version of the Skrull).
The Chitauri faction remained loyal to their Nazi allies right up to the time that the Nazis lost the war. Their mission, however, was actually to prepare the Earth for a Chitauri invasion that was decades to come. Following the fall of the Nazi Party, the Chitauri continued to work from the shadows, eventually infiltrating S.H.I.E.L.D. itself. This arc - seen in Ultimates #1 - #15 - is largely the basis for the MCU storyline of HYDRA taking over S.H.I.E.L.D. from within.
In the Ultimate continuity, we are left to assume that Hitler apparently killed himself just like in real life. Decades later the Chitauri commander, Kleiser, commented to Captain America that Hitler was a fool and never really controlled the war at all.
Earth-199999 (MCU Continuity)
In the MCU continuity, HYDRA is originally a division of the Nazi Party specializing in science & technology and led by the Red Skull. As you stated, we see in the film how it splinters off into its own organization in Captain America: The First Avenger, plus we see in AoS and Captain America: The Winter Soldier what happened to HYDRA after the war.
The ultimate fate of the Nazi Party in the MCU, however, is neither shown on-screen nor mentioned in any supporting materials released so far. Given the "real world" approach taken by the MCU, we may have to assume that following the events of Captain America: The First Avenger, the remainder of the war played out more or less like we know of from history. Under that assumption, Hitler committed suicide as the Allies took Berlin in 1945.
That said, the upcoming "Agent Carter" series will be centered around the formation of the SSR and the remainder of the WWII era. It's logical to assume that we'll get more information on both HYDRA and the Nazis in general from that source.
