Despite growing interest in the mental life of individuals who cannot communicate verbally, objective and non-invasive tests of covert cognition are still sparse. In this study, we assessed the ability of neurotypical children to understand and follow task instructions by measuring neural responses through functional transcranial Doppler ultrasound (fTCD). We recorded blood flow velocity for the two brain hemispheres of twenty children (aged 9 to 12) while they performed either a language task or a visuospatial memory task, on identical visual stimuli. We extracted measures of neural lateralisation for the two tasks separately to investigate lateralisation, and we compared the left-minus-right pattern of activation across tasks to assess task-following. At the group level, we found that neural responses were left-lateralised when children performed the language task, and not when they performed the visuospatial task. However, with statistically robust analyses and controlled paradigms, significant lateralisation in individual children was less frequent than expected from the literature. Nonetheless, the pattern of hemispheric activation for the two tasks allowed us to confirm task-following in the group of participants, as well as in over half of the individuals. This provides a promising avenue for a covert and inexpensive test of children's ability to covertly follow task instructions and perform different mental tasks on identical stimuli.