At the beginning of the movie, why did little William and Duke Hammond assume that Snow White was dead?

Firstly, they left her behind in the middle of a battle. There's a chance that in the midst of the chaos, Snow White will get hurt - whether it's deliberate or accidental. Being a child, it wouldn't take much to deal her a fatal blow. Think about it - someone shoves her out of the way, her head hits a rock and she's probably dead. I'd say the immediate odds are not in her favor.

Assuming she makes it out of the battle, Snow White will inevitably come before Ravenna. She's really too young to escape on her own; her odds of survival alone would not be good. Knowing that Ravenna has just killed the king, it seems very reasonable to assume that she might just kill Snow White then and there. Why she does not is an entirely different question, but again, the odds are not in Snow White's favor.

Barring the immediate threats that would make William and Hammond assume Snow White dead, we have to consider the period of years during her imprisonment. The movie makes it clear that Hammond's castle is a journey of several days away, through some rather nasty territory. We also learn that they are raiding Ravenna, but cannot confront her outright - suggesting that they have rarely been inside Ravenna's castle. Even the two men that get captured and taken inside would never have seen or heard about Snow White - except that she escapes right as the elder man is leaving. Moreover, that man's son was killed as punishment; again, it's not unreasonable to assume that any other men of Hammond's that made it into the castle never came out again alive. Getting news of Snow White out of the castle seems unlikely.

On a further note, Snow White was imprisoned up in a tower. The only people she saw were the occasional other captive put in a cell nearby (like Greta) and her guards. I have a strong suspicion that her guards do not know who she is, a suspicion reinforced by the fact that when Ravenna decides to kill Snow White, she sends her brother personally instead of a guard (remember - the men Ravenna brought in to take over the castle were hers, and would not have known Snow White). If this is true, it would make it even harder for news of Snow White's continued existence to escape the castle. William and Hammond's surprise is not entirely unbelievable.