Choice reaction times are shorter when stimulus and response features are compatible rather than incompatible. Recent studies revealed that spatial compatibility effects in Simon tasks are strongly attenuated when there is temporal overlap with a different high-priority task. In contrast, non-spatial variants of the Simon task appear to be unaffected by task overlap. The present study used the lateralized readiness potential (LRP) within a dual task design to elucidate the dynamics underlying these differential effects for a color and a spatial variant of the Simon task. In the color version there was no sign of early response priming by irrelevant stimulus features in the LRP. The color compatibility effect was independent of task overlap and reflected in the LRP onset latency. In contrast, in the spatial version, priming by irrelevant stimulus location showed up and was mirrored by early LRP activation. Response priming and the corresponding Simon effect, however, were present only in case of little temporal overlap with the primary task. The absence of spatial compatibility effects at strong temporal overlap suggests that response conflicts due to stimulus-related priming depend on the availability of processing resources.