C++ class with a template and different template member variables

My apologies if the title does not accurately describe what I am trying to do.

I am working on a node/scene tree system that is based on the principle of a Doubly Linked List. My base class, Node, has the member functions getParent and getChild that can return the respective node above and below it. I have also implemented the subclasses GameObject and Scene that are Nodes with extra members respective to each subclass.

The idea that I'm trying to implement is that I can instantiate a Scene object which is a Node<Scene>, and then set either its child or its parent to some other Node class, Scene subclass, or a GameObject subclass and have access to all of their respective members and functions. This is my implementation at the moment

template<class classType>
class Node
{
    private:
        std::string name;

        //Hypothetical thing I want to do. Parent or child can be a Node of any type
        template<class T>
        Node<T>* parent;
        template<class T>
        Node<T>* child;

    public:
        Node() {
            // Constructor things...
        };


        // Return the correct class type for the child or parent
       
        Node* getChildNode() { return child; };
        Node* getParentNode() { return parent; };

        
        // Setter functions can accept a node of any type

        template<class T>
        void setChildNode(Node<T> *new_child){
            child = new_child;
        };

        template<class T>
        void setParentNode(Node<T> *new_parent){
            parent = new_parent;
        };
}

class Scene : public Node<Scene>
{
    public:
        Scene();
        void foo();
};

class GameObject : public Node<GameObject>
{
    public:
        GameObject();
        void bar();
};

And the classes would hopefully be used like this:

Scene* root = new Scene();

GameObject* platform = new GameObject();

platform->setParentNode(root); //"error: cannot convert ‘Node<Scene>*’ to ‘Node<GameObject>*’ in assignment"

platform->getParentNode().foo();  //Call function specific to a Scene class

Is there a way that I could achieve this functionality with what I currently have?

Templates in C++ are not a replacement for OO. Templates provide compile-time polymorphism; OO run-time. You need the latter, so you'll need a common base class.