Have you seen these charts?
Hats sized for adult women are generally between 20" (50.8 cm) or 21" (53.3 cm), so for a child 18" (45.7 cm) to 19" (48.3 cm) should work (based on the chart measurements).
As for scarf measurements, I am not up on the latest "in" length for kid scarves; however, looking at scarf patterns for kids the average length seems to lie somewhere between 40" (101.6 cm) and 48" (121.9 cm). Even thought I saw a pattern for a 60" long scarf, I wouldn't make a scarf that long for an active child since it might get caught on stray objects or they could trip on it. Width seems to be pretty steady at 5" (12.7 cm).
I know mom did not want little ears to over hear her request, but is it possible for her to take measurements of the little girls' outwear garments (hat and scarf) for measurement basing for the items you plan to make.