This is not an easy question to answer. It's parallel - Why are folktales so dark when they were meant for children? - is much easier to answer. The folktales were the novels of the folk, and they were generally told for adults. They were collected and abridged for publication, and these publications were often pitched to urban children of an increasingly literate people removed from their rural roots. The printed folktales retained some (but not all) of the darkness of the original oral folktales.

Nursery Rhymes are more complex, and the answer, here, must be more nuanced. First, the origin or nursery rhymes is diverse: some were apparently very old and others may be more recent creations that began their life cycles in print rather than with oral roots. It is not, consequently, easy to talk about them as a group (the way one can with the folktale genre).

The second problem is that not all "nursery" rhymes were necessarily intended for children, so again, we need to take each folk song individually to attempt to understand original intent and audience.

Then we need to circle back to the folktale question and what it implies: while folktales were generally told for adults, there wasn't necessarily a clear separation in the European folk mind, separating children and adults. Folktales were often incredibly long - sometimes taking several nights in the telling. They often had sexuality and violence that would regard as inappropriate for children by Europeans at the time. The late night storytelling sessions, the content, the length and complexity all combine to hint at how children were excluded if not by purpose than by content and the nature of performance. That said, one hears accounts of "old grandmothers" telling folktales to children. No doubt, these were abridged, but we see with that evidence something of the fluidity of the genre when placed in a real setting: the folk did not obey the rules and simply did not behave themselves!

Nursery Rhymes are not a strength of mine, but I suspect we are seeing some of the same issues manifesting here: people sang songs; some songs (but not necessarily all) may have been pitched more for children, while children sometimes sang adult songs, repeating content that was dark and that they may or may not have understood. Some songs were probably intended for children, and some of these have dark content. We can view that dark content in one of two ways: the folk did not see the same dividing line separating children from adults; and/or a dark aspect of a nursery rhyme may have been seen as something children would not understand, but that would amuse adults. We can look at this last point as an echo of some children films that have obvious adult content - allusions to sexuality or themes that pass over the heads of children but amuse the adults. When my son was an infant, I would feed him with Stuka Dive Bottles. He really enjoyed seeing the bottle flying overhead and then making a screaming noise as it dived down to its intended target. I assume he did not understand the horribly dark reference, but he sure enjoyed the game.

edited: text changed in response to criticism.