Besides the Jedi, there are other non-clones serving the Galactic Republic in the Clone Wars. Many of them served at the officer level, particularly in the Republic's star fleets. Unless I am mistaken, most of them believe in the Republic's ideals and the force of good that the Jedi Order represents, as well as having a history of working with the Jedi during the war, sometimes before that too. When Order 66 came down, the suddenness of the clones turning on the Jedi might have surprised and overwhelmed these non-clones before they can react to it, but why did they continue to stay in the Imperial Army? The only non-clones I can observe to have joined the infant Rebel Alliance are mostly grunt troops, primarily non-humans, people from Outer Rim worlds and former Separatists. But among the non-clones who fought alongside the Jedi for years, there seem to be a surprisingly low rate of defection in the years that follow. I can understand the notable exceptions such as Tarkin who positively support Order 66, but what of the majority?
Order 66 did not simply "come down" from nowhere, it was given in response to the attempted arrest/assassination of Palpatine by the Jedi. The general public (including the Senate and non-clone members of the Republic military) were not aware that Palpatine is a Sith Lord or that he had played both sides during the Clone Wars -- all they knew is that the Jedi attempted to assassinate him, leaving him very visibly "scarred and deformed". The general public only knew that the Jedi were attempting to take over the Republic by assassinating the legitimately elected Supreme Chancellor, which is treason.1 This is why the Senate approved of the transition from Republic to Empire "with thunderous applause". The transition was justified by Palpatine as follows:
In order to ensure our security and continuing stability, the Republic will be reorganized into the first Galactic Empire, for a safe and secure society which I assure you will last for ten thousand years.
Emperor Palpatine, Star Wars: Episode III Revenge of the Sith
This justification would sound like a good reason for non-clone members of the Republic military to continue working for the Imperial military -- they would simply be ensuring "security and continuing stability", just as before. Furthermore, the failures of the Republican system were made evident by the Clone Wars, so it would have made sense to transition to an empire.
From the perspective of the general public, the suddenness of Order 66 was due entirely to the suddenness of the failed Jedi assassination attempt -- so there was nothing suspicious about it.
The Jedi had been discredited and the Empire was seen as a force for ensuring security and stability, so there was no reason for the majority of non-clone soldiers to leave the service of the Empire.
1Palpatine himself had of course already committed treason against the Republic by working with the Separatists as Darth Sidious -- but, again, the public didn't know that.