This question is quite broad, but of course child-directed speech also exists in Japanese.
You probably know → is said to be both children's speech as well as child-directed speech, but it is very very common in conversations with no involvement of children. Doubling à la "wakie wakie" is also common (especially for one mora words).
Without further ado, here are some common words/expressions used in child-directed speech:
* →
* →
* →
* →
* →
* →
* →
* →
* →
* →
* →
* →
* →