Scalar implicatures - inferences from a weak description ("I ate some of the cookies") that a stronger alternative is true ("I didn't eat all") - are paradigm cases of pragmatic inference. Children's trouble with scalar implicatures is thus an important puzzle for theories of pragmatic development, given their communicative competence in other domains. Previous research has suggested that access to alternatives might be key. Here, we explore children's reaction times in a new paradigm for measuring scalar implicature processing. Alongside failures on scalar implicatures with "some," we replicate previous reports of failures with "none," and find evidence of a speed-accuracy trade-off for both quantifiers. Motivated by these findings, we explore the relationship between accuracy and reaction time with a Drift Diffusion Model. We find evidence consistent with the hypothesis that preschoolers lack access to alternatives for scalar implicature computation, although this set of alternatives may be broader than previously assumed.