I'm converting a structure diagram to a class with multiple classes associated with compositions. I was able to define the initialization values for a nondefault constructor in a structured diagram, but I can't figure out how to do it with a composition association. Is this possible, or am I going to have to use "None" and instantiate it in the parent class constructor?

There's an initialiser section in the constructor. Use that rather than the implementation. Better still, use parts. Composite relationships are old-hat.