The common terms for seeds and fruit often do not correspond to the botanical classifications. But the Nutting Girl was probably collecting hazelnuts, cultivated as filberts and Kentish Cobs. And these do meet the botanical definition of fruit.

From Wikipedia; However, in botany, a fruit is the ripened ovary or carpel that contains seeds, a nut is a type of fruit and not a seed, and a seed is a ripened ovule.

True nuts are produced, for example, by some plant families of the order Fagales;
Family Fagaceae; Beech (Fagus) Chestnut (Castanea) Oak (Quercus) Stone-oak (Lithocarpus) Tanoak (Notholithocarpus)
Family Betulaceae; Hazel, Filbert (Corylus) Hornbeam (Carpinus)