Slot machine TimeLine animation not working

A much simpler tool for the needed animation is PauseTransition.
If you do not want to specify the layout in the fxml you can override layoutChildren() as you did but you'll need to disable automatic layout (layoutChildren()) so the animation can change positions:

import java.util.List;
import javafx.animation.PauseTransition;
import javafx.scene.Node;
import javafx.scene.layout.Pane;
import javafx.util.Duration;

public class ReelPane extends Pane {

    private List<Node> managed;
    private double dy, numberOfCchildren;
    private boolean isAnimating = false;
    private static final double PAUSE = 1;

    @Override
    protected void layoutChildren() {

        if(isAnimating) return;
        managed = getChildren();
        numberOfCchildren = managed.size();
        double y = 0;
        for (Node node : managed) {
            node.setLayoutX(0);
            node.setLayoutY(y);
            dy = Math.max(dy, node.getBoundsInLocal().getHeight()); //dy stores the highest
            y += dy;
        }
    }
    public void spin() {

        isAnimating = true;
        PauseTransition pause = new PauseTransition(Duration.seconds(PAUSE));
        pause.setOnFinished(event ->{
            for (Node node : managed) {

                if(node.getLayoutY()>= (numberOfCchildren -1) * dy){
                    node.setLayoutY(-dy);
                }
                node.setLayoutY(node.getLayoutY() +dy);
            }
            pause.play();
        });

        pause.play();
    }
}

Consider disabling the button while spinning:

import javafx.fxml.FXML;
import javafx.scene.control.Button;

public class Controller {

    @FXML
    ReelPane reel;
    @FXML
    Button spin;

    public void initialize() {

        spin.setOnAction(event -> {
            reel.spin();
            spin.setDisable(true);
        });
    }
}

To make the code mre here is the slot.fxml using publicly available resources:

<?import javafx.scene.control.Button?>
<?import javafx.scene.image.*?>
<?import javafx.scene.layout.Pane?>
<?import fx_tests.ReelPane?>

<Pane 
      xmlns:fx="http://javafx.com/fxml/1"
      xmlns="http://javafx.com/javafx"
      fx:controller="fx_tests.sample.Controller">

    <ReelPane fx:id="reel">

        <ImageView fitHeight="100" fitWidth="100">
            <Image url="https://cdn3.iconfinder.com/data/icons/softwaredemo/PNG/128x128/Box_Green.png"/>
        </ImageView>

       <ImageView fitHeight="100" fitWidth="100" >
            <Image url="https://cdn3.iconfinder.com/data/icons/softwaredemo/PNG/128x128/Box_Red.png"/>
        </ImageView>

        <ImageView fitHeight="100" fitWidth="100">
            <Image url="https://cdn3.iconfinder.com/data/icons/softwaredemo/PNG/128x128/Box_Yellow.png"/>
        </ImageView>

        <ImageView fitHeight="100" fitWidth="100" >
           <Image url="https://cdn3.iconfinder.com/data/icons/softwaredemo/PNG/128x128/Box_Blue.png"/>
        </ImageView>

        <ImageView fitHeight="100" fitWidth="100">
            <Image url="https://cdn3.iconfinder.com/data/icons/softwaredemo/PNG/128x128/Box_Orange.png"/>
        </ImageView>

    </ReelPane>

    <Button fx:id="spin" text="SPIN" layoutX="20." layoutY="550.0" />

</Pane>

A test application:

import javafx.application.Application;
import javafx.fxml.FXMLLoader;
import javafx.scene.Parent;
import javafx.scene.Scene;
import javafx.stage.Stage;

public class Main extends Application  {

    @Override
    public void start(Stage primaryStage) throws Exception {
        Parent root =
        FXMLLoader.load(getClass().getResource("slot.fxml"));
        primaryStage.setScene(new Scene(root, 200, 600));
        primaryStage.show();

}
    public static void main(String[] args) {
        launch(args);
    }
}