Chestnuts are nuts both botanically and culinarily. Chestnuts are each a whole fruit formed from its own ovary. What you get in the store is the whole thing, with nuts’ characteristic hard shell comprising the chestnut’s entire pericarp. This therefore makes it a botanical nut. Interestingly, they also grow inside of spiked pods in clusters, but this is not part of the fruit itself.
However, you are unlikely to care about the scientific specifics of chestnuts’ botanical divisions. You should only really be concerned about how people eat them since this in the only context you are likely to refer to chestnuts, even in Christmas songs. Chestnuts are eaten as nuts. Therefore, for culinarily purposes, they are nuts.